Appendix: Log-Likelihood Equations: Difference between revisions

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This appendix covers the log-likelihood functions and their associated partial derivatives for most of the distributions available in Weibull++. These distributions are discussed in more detail in Chapters 6 through 10.  
{{Template:LDABOOK|Appendix D|Log-Likelihood Equations}}
 
This appendix covers the log-likelihood functions and their associated partial derivatives for most of the distributions available in Weibull++. These distributions are discussed in more detail in the chapter for each distribution.  
===Weibull Log-Likelihood Functions and their Partials===
===Weibull Log-Likelihood Functions and their Partials===
====The Two-Parameter Weibull====
====The Two-Parameter Weibull====
This log-likelihood function is composed of three summation portions:
This log-likelihood function is composed of three summation portions:


<math>\begin{align}
::<math>\begin{align}
  \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{\eta } \right)}^{\beta }}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{\eta } \right)}^{\beta }} \\  
  \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{\eta } \right)}^{\beta }}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{\eta } \right)}^{\beta }} \\  
   & \text{  }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}} \right]   
   & \text{  }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}} \right]   
\end{align}</math>
\end{align}\,\!</math>


where:
where:


<math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
 
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
<math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
:*<math>\beta \,\!</math> is the Weibull shape parameter (unknown a priori, the first of two parameters to be found)
 
:*<math>\eta \,\!</math> is the Weibull scale parameter (unknown a priori, the second of two parameters to be found)
<math>\beta </math> is the Weibull shape parameter (unknown a priori, the first of two parameters to be found)
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
 
:*<math>S\,\!</math> is the number of groups of suspension data points
<math>\eta </math> is the Weibull scale parameter (unknown a priori, the second of two parameters to be found)
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in <math>{{i}^{th}}\,\!</math> group of suspension data points
 
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
<math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
:*<math>FI\,\!</math> is the number of interval failure data groups
 
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in <math>{{i}^{th}}\,\!</math> group of data intervals
<math>S</math> is the number of groups of suspension data points
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
 
:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval
<math>N_{i}^{\prime }</math> is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points
 
<math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group
 
<math>FI</math> is the number of interval failure data groups
 
<math>N_{i}^{\prime \prime }</math> is the number of intervals in <math>{{i}^{th}}</math> group of data intervals


<math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math>  interval
For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.\,\!</math>


• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval
The solution will be found by solving for a pair of parameters <math>\left( \widehat{\beta },\widehat{\eta } \right)\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial \eta }=0.\,\!</math> It should be noted that other methods can also be used, such as direct maximization of the likelihood function, without having to compute the derivatives.


For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.</math>


The solution will be found by solving for a pair of parameters <math>\left( \widehat{\beta },\widehat{\eta } \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0</math> and <math>\tfrac{\partial \Lambda }{\partial \eta }=0.</math> It should be noted that other methods can also be used, such as direct maximization of the likelihood function, without having to compute the derivatives.
::<math>\begin{align}
 
 
<math>\begin{align}
   \frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}}{\eta } \right) \\  
   \frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}}{\eta } \right) \\  
   & -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{\eta } \right)-\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{\eta } \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{\eta } \right) \\  
   & -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{\eta } \right)-\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{\eta } \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{\eta } \right) \\  
   & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right){{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}+{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right){{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}}   
   & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right){{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}+{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right){{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}}   
\end{align}</math>
\end{align}\,\!</math>






<math>\begin{align}
::<math>\begin{align}
   \frac{\partial \Lambda }{\partial \eta }= & \frac{-\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta }} \\  
   \frac{\partial \Lambda }{\partial \eta }= & \frac{-\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta }} \\  
   & +\frac{\beta }{\eta }\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{\eta } \right)}^{\beta }} \\  
   & +\frac{\beta }{\eta }\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{\eta } \right)}^{\beta }} \\  
   & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{\beta }{\eta } \right){{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-\left( \tfrac{\beta }{\eta } \right){{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}}   
   & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{\beta }{\eta } \right){{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-\left( \tfrac{\beta }{\eta } \right){{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}}   
\end{align}</math>
\end{align}\,\!</math>


====  The Three-Parameter Weibull====
====  The Three-Parameter Weibull====
This log-likelihood function is again composed of three summation portions:
This log-likelihood function is again composed of three summation portions:


<math>\begin{align}
::<math>\begin{align}
  & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }} \\  
  \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }} \\  
&  \\  
  &  \\  
& +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}} \right]   
  & +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}} \right]   
\end{align}</math>
\end{align}\,\!</math>


where,
where:
 
• <math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
 
• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
 
• <math>\beta </math> is the Weibull shape parameter (unknown a priori, the first of three parameters to be found)
 
• <math>\eta </math> is the Weibull scale parameter (unknown a priori, the second of three parameters to be found)
 
• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
 
• <math>\gamma </math> is the Weibull location parameter (unknown a priori, the third of three parameters to be found)


<math>S</math> is the number of groups of suspension data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
:*<math>\beta \,\!</math> is the Weibull shape parameter (unknown a priori, the first of three parameters to be found)
:*<math>\eta \,\!</math> is the Weibull scale parameter (unknown a priori, the second of three parameters to be found)
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
:*<math>\gamma \,\!</math> is the Weibull location parameter (unknown a priori, the third of three parameters to be found)
:*<math>S\,\!</math> is the number of groups of suspension data points
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in <math>{{i}^{th}}\,\!</math> group of suspension data points
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
:*<math>FI\,\!</math> is the number of interval data groups
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
:*and <math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval


<math>N_{i}^{\prime }</math> is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points
The solution is found by solving for <math>\left( \widehat{\beta },\widehat{\eta },\widehat{\gamma } \right)\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,\,\!</math> <math>\tfrac{\partial \Lambda }{\partial \eta }=0,\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial \gamma }=0.\,\!</math>


• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group


<math>FI</math> is the number of interval data groups
::<math>\begin{align}
  \frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}-\gamma }{\eta } \right)-\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}-\gamma }{\eta } \right) \\
  & -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right) \\
  & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} \\
  & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} 
\end{align}\,\!</math>


• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals


• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval


• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval
::<math>\begin{align}
  \frac{\partial \Lambda }{\partial \eta }= & \frac{-\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}+\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }}\left( \frac{\beta }{\eta } \right) \\
  & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\tfrac{\beta }{\eta }{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} \\
  & -\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\tfrac{\beta }{\eta }{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} 
\end{align}\,\!</math>


The solution is found by solving for <math>\left( \widehat{\beta },\widehat{\eta },\widehat{\gamma } \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math> <math>\tfrac{\partial \Lambda }{\partial \eta }=0,</math> and <math>\tfrac{\partial \Lambda }{\partial \gamma }=0.</math>




<math>\begin{align}
  & \frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}-\gamma }{\eta } \right)-\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}-\gamma }{\eta } \right) \\
&  & -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right) \\
&  & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} \\
&  & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} 
\end{align}</math>


::<math>\begin{align}
  \frac{\partial \Lambda }{\partial \gamma }= & \left( 1-\beta  \right)\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,\left( \frac{{{N}_{i}}}{{{T}_{i}}-\gamma } \right)+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}\left( \frac{\beta }{{{T}_{i}}-\gamma } \right) \\
  & +\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }}\left( \frac{\beta }{T_{i}^{\prime }-\gamma } \right) \\
  & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\tfrac{\beta }{T_{Li}^{\prime \prime }-\gamma }{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-\tfrac{\beta }{T_{Ri}^{\prime \prime }-\gamma }{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} 
\end{align}\,\!</math>




<math>\begin{align}
It should be pointed out that the solution to the three-parameter Weibull via MLE is not always stable and can collapse if <math>\beta \sim 1.\,\!</math> In estimating the true MLE of the three-parameter Weibull distribution, two difficulties arise. The first is a problem of non-regularity and the second is the parameter divergence problem, as discussed in Hirose [[Appendix:_Life_Data_Analysis_References|[14]]].
  & \frac{\partial \Lambda }{\partial \eta }= & \frac{-\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}+\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }}\left( \frac{\beta }{\eta } \right) \\
&  & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\tfrac{\beta }{\eta }{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} \\
&  & -\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\tfrac{\beta }{\eta }{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} 
\end{align}</math>


Non-regularity occurs when <math>\beta \le 2.\,\!</math> In general, there are no MLE solutions in the region of <math>0<\beta <1.\,\!</math> When <math>1<\beta <2,\,\!</math> MLE solutions exist but are not asymptotically normal, as discussed in Hirose [[Appendix:_Life_Data_Analysis_References|[14]]]. In the case of non-regularity, the solution is treated anomalously.


 
Weibull++ attempts to find a solution in all of the regions using a variety of methods, but the user should be forewarned that not all possible data can be addressed. Thus, some solutions using MLE for the three-parameter Weibull will fail when the algorithm has reached predefined limits or fails to converge. In these cases, the user can change to the non-true MLE approach (in Weibull++ Application Setup), where <math>\gamma \,\!</math> is estimated using non-linear regression. Once <math>\gamma \,\!</math> is obtained, the MLE estimates of <math>\widehat{\beta }\,\!</math> and <math>\widehat{\eta }\,\!</math> are computed using the transformation <math>T_{i}^{\prime }=({{T}_{i}}-\gamma ).\,\!</math>
 
<math>\begin{align}
  & \frac{\partial \Lambda }{\partial \gamma }= & \left( 1-\beta  \right)\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,\left( \frac{{{N}_{i}}}{{{T}_{i}}-\gamma } \right)+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}\left( \frac{\beta }{{{T}_{i}}-\gamma } \right) \\
&  & +\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }}\left( \frac{\beta }{T_{i}^{\prime }-\gamma } \right) \\
&  & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\tfrac{\beta }{T_{Li}^{\prime \prime }-\gamma }{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-\tfrac{\beta }{T_{Ri}^{\prime \prime }-\gamma }{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} 
\end{align}</math>
 
 
It should be pointed out that the solution to the three-parameter Weibull via MLE is not always stable and can collapse if <math>\beta \sim 1.</math> In estimating the true MLE of the three-parameter Weibull distribution, two difficulties arise. The first is a problem of non-regularity and the second is the parameter divergence problem [14].
Non-regularity occurs when <math>\beta \le 2.</math> In general, there are no MLE solutions in the region of <math>0<\beta <1.</math> When <math>1<\beta <2,</math> MLE solutions exist but are not asymptotically normal [14]. In the case of non-regularity, the solution is treated anomalously.
 
Weibull++ attempts to find a solution in all of the regions using a variety of methods, but the user should be forewarned that not all possible data can be addressed. Thus, some solutions using MLE for the three-parameter Weibull will fail when the algorithm has reached predefined limits or fails to converge. In these cases, the user can change to the non-true MLE approach (in Weibull++ User Setup), where <math>\gamma </math> is estimated using non-linear regression. Once <math>\gamma </math> is obtained, the MLE estimates of <math>\widehat{\beta }</math> and <math>\widehat{\eta }</math> are computed using the transformation <math>T_{i}^{\prime }=({{T}_{i}}-\gamma ).</math>


===  Exponential Log-Likelihood Functions and their Partials===
===  Exponential Log-Likelihood Functions and their Partials===
Line 128: Line 108:
This log-likelihood function is composed of three summation portions:
This log-likelihood function is composed of three summation portions:


<math>\ln (L)=\Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-\lambda T_{Li}^{\prime \prime }}}-{{e}^{-\lambda T_{Ri}^{\prime \prime }}} \right]</math>
::<math>\ln (L)=\Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-\lambda T_{Li}^{\prime \prime }}}-{{e}^{-\lambda T_{Ri}^{\prime \prime }}} \right]\,\!</math>


where:
where:


<math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
 
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
<math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
:*<math>\lambda \,\!</math> is the failure rate parameter (unknown a priori, the only parameter to be found)
 
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
<math>\lambda </math> is the failure rate parameter (unknown a priori, the only parameter to be found)
:*<math>S\,\!</math> is the number of groups of suspension data points
 
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points
<math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
 
:*<math>FI\,\!</math> is the number of interval data groups
<math>S</math> is the number of groups of suspension data points
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval


<math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points
The solution will be found by solving for a parameter <math>\widehat{\lambda }\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial \lambda }=0.\,\!</math> Note that for <math>FI=0\,\!</math> there exists a closed form solution.


• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group
::<math>\begin{align}
 
   \frac{\partial \Lambda }{\partial \lambda }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{1}{\lambda }-{{T}_{i}} \right)-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }T_{i}^{\prime } \\  
• <math>FI</math> is the number of interval data groups
  & -\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\left[ \frac{T_{Li}^{\prime \prime }{{e}^{-\lambda T_{Li}^{\prime \prime }}}-T_{Ri}^{\prime \prime }{{e}^{-\lambda T_{Ri}^{\prime \prime }}}}{{{e}^{-\lambda T_{Li}^{\prime \prime }}}-{{e}^{-\lambda T_{Ri}^{\prime \prime }}}} \right]   
 
\end{align}\,\!</math>
• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals
 
• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval
 
• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the  <math>{{i}^{th}}</math> interval
 
The solution will be found by solving for a parameter <math>\widehat{\lambda }</math> so that <math>\tfrac{\partial \Lambda }{\partial \lambda }=0.</math> Note that for <math>FI=0</math> there exists a closed form solution.
 
<math>\begin{align}
   & \frac{\partial \Lambda }{\partial \lambda }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{1}{\lambda }-{{T}_{i}} \right)-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }T_{i}^{\prime } \\  
& -\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\left[ \frac{T_{Li}^{\prime \prime }{{e}^{-\lambda T_{Li}^{\prime \prime }}}-T_{Ri}^{\prime \prime }{{e}^{-\lambda T_{Ri}^{\prime \prime }}}}{{{e}^{-\lambda T_{Li}^{\prime \prime }}}-{{e}^{-\lambda T_{Ri}^{\prime \prime }}}} \right]   
\end{align}</math>


====  The Two-Parameter Exponential====
====  The Two-Parameter Exponential====
Line 165: Line 135:




<math>\begin{align}
::<math>\begin{align}
   & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda \left( {{T}_{i}}-\gamma  \right)}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda \left( T_{i}^{\prime }-\gamma  \right) \\  
   & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda \left( {{T}_{i}}-\gamma  \right)}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda \left( T_{i}^{\prime }-\gamma  \right) \\  
  &  & \ \ +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-\lambda \left( T_{Li}^{\prime \prime }-\gamma  \right)}}-{{e}^{-\lambda \left( T_{Ri}^{\prime \prime }-\gamma  \right)}} \right],   
  &  & \ \ +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-\lambda \left( T_{Li}^{\prime \prime }-\gamma  \right)}}-{{e}^{-\lambda \left( T_{Ri}^{\prime \prime }-\gamma  \right)}} \right],   
\end{align}</math>
\end{align}\,\!</math>


where,
where:


<math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
:*<math>\lambda \,\!</math> is the failure rate parameter (unknown a priori, the first of two parameters to be found)
:*<math>\gamma \,\!</math> is the location parameter (unknown a priori, the second of two parameters to be found)
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
:*<math>S\,\!</math> is the number of groups of suspension data points
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
:*<math>FI\,\!</math> is the number of interval data groups
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval


<math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
To find the two-parameter solution, look at the partial derivatives <math>\tfrac{\partial \Lambda }{\partial \lambda }</math> and <math>\tfrac{\partial \Lambda }{\partial \gamma}</math>:


• <math>\lambda </math> is the failure rate parameter (unknown a priori, the first of two parameters to be found)
::<math>\begin{align}
 
• <math>\gamma </math> is the location parameter (unknown a priori, the second of two parameters to be found)
 
• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
 
• <math>S</math> is the number of groups of suspension data points
 
• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points
 
• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group
 
• <math>FI</math> is the number of interval data groups
 
• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals
 
• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval
 
• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval
 
The two-parameter solution will be found by solving for a pair of parameters (<math>\widehat{\lambda },\widehat{\gamma }),</math> such that <math>\tfrac{\partial \Lambda }{\partial \lambda }=0,\tfrac{\partial \Lambda }{\partial \gamma }=0.</math> For the one-parameter case, solve for <math>\tfrac{\partial \Lambda }{\partial \lambda }=0.</math>
 
<math>\begin{align}
   \frac{\partial \Lambda }{\partial \lambda }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left[ \frac{1}{\lambda }-\left( {{T}_{i}}-\gamma  \right) \right] \\  
   \frac{\partial \Lambda }{\partial \lambda }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left[ \frac{1}{\lambda }-\left( {{T}_{i}}-\gamma  \right) \right] \\  
   & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\left( T_{i}^{\prime }-\gamma  \right) \\  
   & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\left( T_{i}^{\prime }-\gamma  \right) \\  
   & -\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\left[ \frac{\left( T_{Li}^{\prime \prime }-\gamma  \right){{e}^{-\lambda \left( T_{Li}^{\prime \prime }-{{\gamma }_{0}} \right)}}-\left( T_{Ri}^{\prime \prime }-\gamma  \right){{e}^{-\lambda \left( T_{Ri}^{\prime \prime }-\gamma  \right)}}}{{{e}^{-\lambda \left( T_{Li}^{\prime \prime }-\gamma  \right)}}-{{e}^{-\lambda \left( T_{Ri}^{\prime \prime }-\gamma  \right)}}} \right]   
   & -\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\left[ \frac{\left( T_{Li}^{\prime \prime }-\gamma  \right){{e}^{-\lambda \left( T_{Li}^{\prime \prime }-{{\gamma }_{0}} \right)}}-\left( T_{Ri}^{\prime \prime }-\gamma  \right){{e}^{-\lambda \left( T_{Ri}^{\prime \prime }-\gamma  \right)}}}{{{e}^{-\lambda \left( T_{Li}^{\prime \prime }-\gamma  \right)}}-{{e}^{-\lambda \left( T_{Ri}^{\prime \prime }-\gamma  \right)}}} \right]   
\end{align}</math>
\end{align}\,\!</math>


and:
and


<math>\frac{\partial \Lambda }{\partial \gamma }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\lambda +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\lambda </math>
::<math>\begin{align}\frac{\partial \Lambda }{\partial \gamma }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\lambda +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\lambda \,\!\end{align}</math>.


Examination of Eqn. (expll1) will reveal that:
From here we see that <math>\frac{\partial \Lambda }{\partial \gamma}</math> is a positive, constant function of <math>\gamma</math>.  As alluded to in the chapter on the exponential distribution, this implies that the log-likelihood function <math>\Lambda</math> is, for fixed <math>\lambda</math>, an increasing function of <math>\gamma</math>.  Thus the MLE for <math>\gamma</math> is its largest possible value <math>T_1</math>. Therefore, to find the full MLE solution <math>(\widehat{\lambda },\widehat{\gamma})</math> for the two-parameter exponential distribution, one should set <math>\gamma</math> equal to the first failure time and then find (numerically) a <math>\lambda</math> such that <math>\tfrac{\partial \Lambda}{\partial \lambda} = 0</math>.


<math>\frac{\partial \Lambda }{\partial \gamma }=\left( \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ \ +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime } \right)\lambda \equiv 0</math>
The 3D Plot utility in Weibull++ further illustrates this behavior of the log-likelihood function, as shown next:


or Eqn. (expll2) will be equal to zero only if either:
[[image: appendixc__127.gif|center|350px]]
 
<math>\lambda =0</math>
 
or:
 
<math>\left( \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ \ +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime } \right)=0</math>
 
This is an unwelcome fact, alluded to earlier in the chapter, that essentially indicates that there is no realistic solution for the two-parameter MLE for exponential. The above equations indicate that there is no non-trivial MLE solution that satisfies both <math>\tfrac{\partial \Lambda }{\partial \lambda }=0,\tfrac{\partial \Lambda }{\partial \gamma }=0.</math>
It can be shown that the best solution for <math>\gamma ,</math> satisfying the constraint that <math>\gamma \le {{T}_{1}}</math> is <math>\gamma ={{T}_{1}}.</math> To then solve for the two-parameter exponential distribution via MLE, one can set  equal to the first time-to-failure, and then find a <math>\lambda </math> such that <math>\tfrac{\partial \Lambda }{\partial \lambda }=0.</math>
 
Using this methodology, a maximum can be achieved along the <math>\lambda </math>-axis, and a local maximum along the <math>\gamma </math>-axis at <math>\gamma ={{T}_{1}}</math>, constrained by the fact that <math>\gamma \le {{T}_{1}}</math>. The 3D Plot utility in Weibull++ illustrates this behavior of the log-likelihood function, as shown next:
 
<math></math>


===  Normal Log-Likelihood Functions and their Partials===
===  Normal Log-Likelihood Functions and their Partials===
Line 231: Line 177:




<math>\begin{align}
::<math>\begin{align}
   \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{\sigma }\phi \left( \frac{{{T}_{i}}-\mu }{\sigma } \right) \right] \\  
   \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{\sigma }\phi \left( \frac{{{T}_{i}}-\mu }{\sigma } \right) \right] \\  
   & +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{^{\prime }}\ln \left[ 1-\Phi \left( \frac{T_{i}^{^{\prime }}-\mu }{\sigma } \right) \right] \\  
   & +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{^{\prime }}\ln \left[ 1-\Phi \left( \frac{T_{i}^{^{\prime }}-\mu }{\sigma } \right) \right] \\  
   & \text{ }+\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{^{\prime \prime }}\ln \left[ \Phi \left( \frac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma } \right)-\Phi \left( \frac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma } \right) \right]   
   & \text{ }+\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{^{\prime \prime }}\ln \left[ \Phi \left( \frac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma } \right)-\Phi \left( \frac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma } \right) \right]   
\end{align}</math>
\end{align}\,\!</math>




where:
where:


<math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
:*<math>\mu \,\!</math> is the mean parameter (unknown a priori, the first of two parameters to be found)
:*<math>\sigma \,\!</math> is the standard deviation parameter (unknown a priori, the second of two parameters to be found)
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
:*<math>S\,\!</math> is the number of groups of suspension data points
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
:*<math>{{F}_{i}}\,\!</math> is the number of interval data groups
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval


<math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
The solution will be found by solving for a pair of parameters <math>\left( {{\mu }_{0}},{{\sigma }_{0}} \right)\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial \mu }=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial \sigma }=0.\,\!</math>


• <math>\mu </math> is the mean parameter (unknown a priori, the first of two parameters to be found)


• <math>\sigma </math> is the standard deviation parameter (unknown a priori, the second of two parameters to be found)
::<math>\begin{align}
 
• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
 
• <math>S</math> is the number of groups of suspension data points
 
• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points
 
• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group
 
• <math>{{F}_{i}}</math> is the number of interval data groups
 
• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals
 
• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval
 
• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval
 
The solution will be found by solving for a pair of parameters <math>\left( {{\mu }_{0}},{{\sigma }_{0}} \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial \mu }=0</math> and <math>\tfrac{\partial \Lambda }{\partial \sigma }=0.</math>
 
 
<math>\begin{align}
   \frac{\partial \Lambda }{\partial \mu }= & \frac{1}{{{\sigma }^{2}}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}({{T}_{i}}-\mu ) \\  
   \frac{\partial \Lambda }{\partial \mu }= & \frac{1}{{{\sigma }^{2}}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}({{T}_{i}}-\mu ) \\  
   & +\frac{1}{\sigma }\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)}{1-\Phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)} \\  
   & +\frac{1}{\sigma }\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)}{1-\Phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)} \\  
   & -\frac{1}{\sigma }\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}{\Phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\Phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}   
   & -\frac{1}{\sigma }\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}{\Phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\Phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}   
\end{align}</math>
\end{align}\,\!</math>






<math>\begin{align}
::<math>\begin{align}
   \frac{\partial \Lambda }{\partial \sigma }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( {{T}_{i}}-\mu  \right)}^{2}}}{{{\sigma }^{3}}}-\frac{1}{\sigma } \right) \\  
   \frac{\partial \Lambda }{\partial \sigma }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( {{T}_{i}}-\mu  \right)}^{2}}}{{{\sigma }^{3}}}-\frac{1}{\sigma } \right) \\  
   & +\frac{1}{\sigma }\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)}{1-\Phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)} \\  
   & +\frac{1}{\sigma }\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)}{1-\Phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)} \\  
   & -\frac{1}{\sigma }\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}{\Phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\Phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}   
   & -\frac{1}{\sigma }\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}{\Phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\Phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}   
\end{align}</math>
\end{align}\,\!</math>


where:
where:


<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>
::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{{{x}^{2}}}{2}}}\,\!</math>


and:
and:


<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>
::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt\,\!</math>




====  Complete Data====
====  Complete Data====
Note that for the normal distribution, and in the case of complete data only (as was shown in Chapter 3), there exists a closed-form solution for both of the parameters or:
Note that for the normal distribution, and in the case of complete data only (as was shown in [[Basic Statistical Background]]), there exists a closed-form solution for both of the parameters or:




<math>\widehat{\mu }=\widehat{{\bar{T}}}=\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{T}_{i}}</math>
::<math>\widehat{\mu }=\widehat{{\bar{T}}}=\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{T}_{i}}\,\!</math>


and:
and:


<math>\begin{align}
::<math>\begin{align}
   \hat{\sigma }_{T}^{2}= & \frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{({{T}_{i}}-\bar{T})}^{2}} \\  
   \hat{\sigma }_{T}^{2}= & \frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{({{T}_{i}}-\bar{T})}^{2}} \\  
   {{{\hat{\sigma }}}_{T}}= & \sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{({{T}_{i}}-\bar{T})}^{2}}}   
   {{{\hat{\sigma }}}_{T}}= & \sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{({{T}_{i}}-\bar{T})}^{2}}}   
\end{align}</math>
\end{align}\,\!</math>


===  Lognormal Log-Likelihood Functions and their Partials===
===  Lognormal Log-Likelihood Functions and their Partials===
Line 307: Line 242:




<math>\begin{align}
::<math>\begin{align}
   \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right] \\  
   \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{T}_{i}} {{\sigma }_{{{T}'}}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right] \\  
   & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right] \\  
   & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right] \\  
   & \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ \Phi \left( \frac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \frac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right]   
   & \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ \Phi \left( \frac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \frac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right]   
\end{align}</math>
\end{align}\,\!</math>


where:
where:


<math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
 
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
<math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
:*<math>{\mu }'\,\!</math> is the mean of the natural logarithms of the times-to-failure (unknown a priori, the first of two parameters to be found)
 
:*<math>{{\sigma }_{{{T}'}}}\,\!</math> is the standard deviation of the natural logarithms of the times-to-failure (unknown a priori, the second of two parameters to be found)
<math>{\mu }'</math> is the mean of the natural logarithms of the times-to-failure (unknown a priori, the first of two parameters to be found)
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
 
:*<math>S\,\!</math> is the number of groups of suspension data points
<math>{{\sigma }_{{{T}'}}}</math> is the standard deviation of the natural logarithms of the times-to-failure (unknown a priori, the second of two parameters to be found)
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points
 
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
<math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
:*<math>FI\,\!</math> is the number of interval data groups
 
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals
<math>S</math> is the number of groups of suspension data points
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
 
:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval
<math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points
 
<math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group


<math>FI</math> is the number of interval data groups
The solution will be found by solving for a pair of parameters <math>\left( {\mu }',{{\sigma }_{{{T}'}}} \right)\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial {\mu }'}=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0\,\!</math>:


• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals


• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval


• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval
::<math>\begin{align}
 
   \frac{\partial \Lambda }{\partial {\mu }'}= & \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,N_{i}(\ln ({{T}_{i}})-{\mu }') \\  
The solution will be found by solving for a pair of parameters <math>\left( {\mu }',{{\sigma }_{{{T}'}}} \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial {\mu }'}=0</math> and <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0</math>:
 
 
 
<math>\begin{align}
   \frac{\partial \Lambda }{\partial {\mu }'}= & \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})-{\mu }') \\  
   & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} \\  
   & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} \\  
   & \ \ -\underset{i=1}{\overset{FI}{\mathop \sum }}\,\frac{N_{i}^{\prime \prime }}{\sigma }\frac{\phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}   
   & \ \ -\underset{i=1}{\overset{FI}{\mathop \sum }}\,\frac{N_{i}^{\prime \prime }}{\sigma }\frac{\phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}   
\end{align}</math>
\end{align}\,\!</math>






<math></math>
::<math>\begin{align}
  \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,N_{i}\left( \frac{{{\left( \ln ({{T}_{i}})-{\mu }' \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) \\
  & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} \\
  & -\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} 
\end{align}\,\!</math>


where:
where:


<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>
::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{{{x}^{2}}}{2}}}\,\!</math>


and:
and:


<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>
::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt\,\!</math>


===  Mixed Weibull Log-Likelihood Functions and their Partials===
===  Mixed Weibull Log-Likelihood Functions and their Partials===
The log-likelihood function  (without the constant) is composed of three summation portions:  
The log-likelihood function  (without the constant) is composed of three summation portions:  
<math>\begin{align}
  \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})-{\mu }' \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) \\
  & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} \\
  & -\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} 
\end{align}</math>


<math>\begin{align}
::<math>\begin{align}
   \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}\frac{{{\beta }_{k}}}{{{\eta }_{k}}}{{\left( \frac{{{T}_{i}}}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}-1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right] \\  
   \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}\frac{{{\beta }_{k}}}{{{\eta }_{k}}}{{\left( \frac{{{T}_{i}}}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}-1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right] \\  
   & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}{{e}^{-{{\left( \tfrac{T_{i}^{\prime }}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right] \\  
   & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}{{e}^{-{{\left( \tfrac{T_{i}^{\prime }}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right] \\  
   & \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}\frac{{{\beta }_{k}}}{{{\eta }_{k}}}{{\left( \frac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{{\eta }_{k}}} \right)}^{{{\beta }_{k}}-1}}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right]   
   & \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}\frac{{{\beta }_{k}}}{{{\eta }_{k}}}{{\left( \frac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{{\eta }_{k}}} \right)}^{{{\beta }_{k}}-1}}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right]   
\end{align}</math>
\end{align}\,\!</math>
   
   
where:
where:


<math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
 
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
<math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
:*<math>Q\,\!</math> is the number of subpopulations
 
:*<math>{{\rho }_{k}}\,\!</math> is the proportionality of the <math>{{k}^{th}}\,\!</math> subpopulation (unknown a priori, the first set of three sets of parameters to be found)
<math>Q</math> is the number of subpopulations
:*<math>{{\beta }_{k}}\,\!</math> is the Weibull shape parameter of the <math>{{k}^{th}}\,\!</math> subpopulation (unknown a priori, the second set of three sets of parameters to be found)
 
:*<math>{{\eta }_{k}}\,\!</math> is the Weibull scale parameter (unknown a priori, the third set of three sets of parameters to be found)
<math>{{\rho }_{k}}</math> is the proportionality of the <math>{{k}^{th}}</math> subpopulation (unknown a priori, the first set of three sets of parameters to be found)
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
 
:*<math>S\,\!</math> is the number of groups of suspension data points
<math>{{\beta }_{k}}</math> is the Weibull shape parameter of the <math>{{k}^{th}}</math> subpopulation (unknown a priori, the second set of three sets of parameters to be found)
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in <math>{{i}^{th}}\,\!</math> group of suspension data points
 
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
<math>{{\eta }_{k}}</math> is the Weibull scale parameter (unknown a priori, the third set of three sets of parameters to be found)
:*<math>FI\,\!</math> is the number of groups of interval data points
 
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in <math>{{i}^{th}}\,\!</math> group of data intervals
<math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
 
:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval
<math>S</math> is the number of groups of suspension data points
 
<math>N_{i}^{\prime }</math> is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points
 
<math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group
 
<math>FI</math> is the number of groups of interval data points
 
<math>N_{i}^{\prime \prime }</math> is the number of intervals in <math>{{i}^{th}}</math> group of data intervals
 
<math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval
 
• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval


The solution will be found by solving for a group of parameters:
The solution will be found by solving for a group of parameters:


<math>\left( \widehat{{{\rho }_{1,}}}\widehat{{{\beta }_{1}}},\widehat{{{\eta }_{1}}},\widehat{{{\rho }_{2,}}}\widehat{{{\beta }_{2}}},\widehat{{{\eta }_{2}}},...,\widehat{{{\rho }_{Q,}}}\widehat{{{\beta }_{Q}}},\widehat{{{\eta }_{Q}}} \right)</math>
::<math>\left( \widehat{{{\rho }_{1,}}}\widehat{{{\beta }_{1}}},\widehat{{{\eta }_{1}}},\widehat{{{\rho }_{2,}}}\widehat{{{\beta }_{2}}},\widehat{{{\eta }_{2}}},...,\widehat{{{\rho }_{Q,}}}\widehat{{{\beta }_{Q}}},\widehat{{{\eta }_{Q}}} \right)\,\!</math>


so that:
so that:


<math>\begin{align}
::<math>\begin{align}
   \frac{\partial \Lambda }{\partial {{\rho }_{1}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{1}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{1}}}=0 \\  
   \frac{\partial \Lambda }{\partial {{\rho }_{1}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{1}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{1}}}=0 \\  
   \frac{\partial \Lambda }{\partial {{\rho }_{2}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{2}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{2}}}=0 \\  
   \frac{\partial \Lambda }{\partial {{\rho }_{2}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{2}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{2}}}=0 \\  
Line 418: Line 327:
   \frac{\partial \Lambda }{\partial {{\rho }_{Q-1}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{Q-1}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{Q-1}}}=0 \\  
   \frac{\partial \Lambda }{\partial {{\rho }_{Q-1}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{Q-1}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{Q-1}}}=0 \\  
   \frac{\partial \Lambda }{\partial {{\beta }_{Q}}}= & 0,\text{ and }\frac{\partial \Lambda }{\partial {{\eta }_{Q}}}=0   
   \frac{\partial \Lambda }{\partial {{\beta }_{Q}}}= & 0,\text{ and }\frac{\partial \Lambda }{\partial {{\eta }_{Q}}}=0   
\end{align}</math>
\end{align}\,\!</math>


===  Logistic Log-Likelihood Functions and their Partials===
===  Logistic Log-Likelihood Functions and their Partials===
This log-likelihood function is composed of three summation portions:
This log-likelihood function is composed of three summation portions:


<math>\begin{align}
::<math>\begin{align}
   & \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}{\sigma {{(1+{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}})}^{2}}} \right)-\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln (1+{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}) \\  
   \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}{\sigma {{(1+{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}})}^{2}}} \right)-\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln (1+{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}) \\  
& & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( \frac{1}{1+{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}-\frac{1}{1+{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}} \right)   
  & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( \frac{1}{1+{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}-\frac{1}{1+{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}} \right)   
\end{align}</math>
\end{align}\,\!</math>


where:
where:


<math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
 
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
<math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
:*<math>\mu \,\!</math> is the logistic shape parameter (unknown a priori, the first of two parameters to be found)
 
:*<math>\eta \,\!</math> is the logistic scale parameter (unknown a priori, the second of two parameters to be found)
<math>\mu </math> is the logistic shape parameter (unknown a priori, the first of two parameters to be found)
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
 
:*<math>S\,\!</math> is the number of groups of suspension data points
<math>\eta </math> is the logistic scale parameter (unknown a priori, the second of two parameters to be found)
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in <math>{{i}^{th}}\,\!</math> group of suspension data points
 
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
<math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
:*<math>FI\,\!</math> is the number of interval failure data group
 
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in <math>{{i}^{th}}\,\!</math> group of data intervals
<math>S</math> is the number of groups of suspension data points
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval


<math>N_{i}^{\prime }</math> is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points
For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.\,\!</math>


<math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group
The solution of the maximum log-likelihood function is found by solving for (<math>\widehat{\mu },\widehat{\sigma })\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial \sigma }=0.\,\!</math>


• <math>FI</math> is the number of interval failure data group


<math>N_{i}^{\prime \prime }</math> is the number of intervals in <math>{{i}^{th}}</math> group of data intervals
::<math>\begin{align}
  \frac{\partial \Lambda }{\partial \mu }= & -\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}} \\
& -\frac{\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\mathop{}_{}^{}}}\,}}\,N_{i}^{^{\prime \prime }}}{\sigma }+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}+\frac{{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}} \right) 
\end{align}\,\!</math>


• <math>T_{Li}^{\prime \prime }</math> is the beginning of the  <math>{{i}^{th}}</math> interval


• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the  <math>{{i}^{th}}</math> interval


For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.</math>


The solution of the maximum log-likelihood function is found by solving for (<math>\widehat{\mu },\widehat{\sigma })</math> so that <math>\tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial \sigma }=0.</math>
::<math>\begin{align}
 
  \frac{\partial \Lambda }{\partial \sigma }= & -\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{T}_{i}}-\mu }{{{\sigma }^{2}}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{\tfrac{{{T}_{i}}-\mu }{\sigma }{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}} \\
 
  & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}} \\  
<math>\begin{align}
& \frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}(\frac{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}+\frac{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}} \\  
  & \frac{\partial \Lambda }{\partial \mu }= & -\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}} \\  
  & -\frac{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}-\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}
&  & -\frac{\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\mathop{}_{}^{}}}\,}}\,N_{i}^{^{\prime \prime }}}{\sigma }+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}+\frac{{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}} \right) 
\end{align}\,\!</math>
\end{align}</math>
 
 
 
 
<math>\begin{align}
  & \frac{\partial \Lambda }{\partial \sigma }= & -\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{T}_{i}}-\mu }{{{\sigma }^{2}}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{\tfrac{{{T}_{i}}-\mu }{\sigma }{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}} \\
&  & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}} \\
&  & \frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}(\frac{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}+\frac{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}} \\
&  & -\frac{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}-\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}) 
\end{align}</math>


===  The Loglogistic Log-Likelihood Functions and their Partials===
===  The Loglogistic Log-Likelihood Functions and their Partials===
This log-likelihood function is composed of three summation portions:
This log-likelihood function is composed of three summation portions:


<math>\begin{align}
::<math>\begin{align}
   \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}}{\sigma t{{(1+{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}})}^{2}}} \right) \\  
   \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}}{\sigma t{{(1+{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}})}^{2}}} \right) \\  
   & -\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln (1+{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}) \\  
   & -\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln (1+{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}) \\  
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( \frac{1}{1+{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}-\frac{1}{1+{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}} \right)   
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( \frac{1}{1+{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}-\frac{1}{1+{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}} \right)   
\end{align}</math>
\end{align}\,\!</math>


where:
where:


<math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
 
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
<math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
:*<math>\mu \,\!</math> is the loglogistic shape parameter (unknown a priori, the first of two parameters to be found)
:*<math>\sigma \,\!</math> is the loglogistic scale parameter (unknown a priori, the second of two parameters to be found)
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
:*<math>S\,\!</math> is the number of groups of suspension data points
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in <math>{{i}^{th}}\,\!</math> group of suspension data points
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
:*<math>FI\,\!</math> is the number of interval failure data groups,
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in <math>{{i}^{th}}\,\!</math> group of data intervals
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval


<math>\mu </math> is the loglogistic shape parameter (unknown a priori, the first of two parameters to be found)
For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.\,\!</math>


<math>\sigma </math> is the loglogistic scale parameter (unknown a priori, the second of two parameters to be found)
The solution of the maximum log-likelihood function is found by solving for (<math>\widehat{\mu },\widehat{\sigma })\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial \sigma }=0.\,\!</math>


• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data


• <math>S</math> is the number of groups of suspension data points
::<math>\begin{align}
 
• <math>N_{i}^{\prime }</math> is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points
 
• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group
 
• <math>FI</math> is the number of interval failure data groups,
 
• <math>N_{i}^{\prime \prime }</math> is the number of intervals in <math>{{i}^{th}}</math> group of data intervals
 
• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval
 
• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval
 
For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.</math>
 
The solution of the maximum log-likelihood function is found by solving for (<math>\widehat{\mu },\widehat{\sigma })</math> so that <math>\tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial \sigma }=0.</math>
 
 
<math>\begin{align}
   \frac{\partial \Lambda }{\partial \mu }= & -\frac{\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\mathop{}_{}^{}}}\,}}\,{{N}_{i}}}{\sigma }+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}} \\  
   \frac{\partial \Lambda }{\partial \mu }= & -\frac{\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\mathop{}_{}^{}}}\,}}\,{{N}_{i}}}{\sigma }+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}} \\  
   & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}}-\frac{{{F}_{I}}}{\sigma } \\  
   & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}}-\frac{{{F}_{I}}}{\sigma } \\  
   & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}+\frac{{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}} \right)   
   & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}+\frac{{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}} \right)   
\end{align}</math>
\end{align}\,\!</math>






<math>\begin{align}
::<math>\begin{align}
   \frac{\partial \Lambda }{\partial \sigma }= & -\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{\ln ({{T}_{i}})-\mu }{{{\sigma }^{2}}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}} \\  
   \frac{\partial \Lambda }{\partial \sigma }= & -\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{\ln ({{T}_{i}})-\mu }{{{\sigma }^{2}}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}} \\  
   & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}} \\  
   & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}} \\  
   & \frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}(\frac{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}+\frac{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}} \\  
   & \frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}(\frac{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}+\frac{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}} \\  
   & -\frac{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}-\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}-{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}})   
   & -\frac{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}-\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}-{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}})   
\end{align}</math>
\end{align}\,\!</math>


===  The Gumbel Log-Likelihood Functions and their Partials===
===  The Gumbel Log-Likelihood Functions and their Partials===
This log-likelihood function is composed of three summation portions:
This log-likelihood function is composed of three summation portions:


<math>\begin{align}
::<math>\begin{align}
   \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }-{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}}}{\sigma } \right) \\  
   \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }-{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}}}{\sigma } \right) \\  
   & -\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln \left( {{e}^{-{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}}} \right) \\  
   & -\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln \left( {{e}^{-{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}}} \right) \\  
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}} \right)   
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}} \right)   
\end{align}</math>
\end{align}\,\!</math>


or:


or
::<math>\begin{align}
 
 
<math>\begin{align}
   \Lambda = & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( \frac{{{T}_{i}}-\mu }{\sigma }-{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}} \right)-\ln (\sigma )\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}} \\  
   \Lambda = & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( \frac{{{T}_{i}}-\mu }{\sigma }-{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}} \right)-\ln (\sigma )\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}} \\  
   & +\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}} \\  
   & +\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}} \\  
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}} \right)   
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}} \right)   
\end{align}</math>
\end{align}\,\!</math>


where:
where:


<math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
:*<math>\mu \,\!</math> is the Gumbel shape parameter (unknown a priori, the first of two parameters to be found)
:*<math>\sigma \,\!</math> is the Gumbel scale parameter (unknown a priori, the second of two parameters to be found)
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
:*<math>S\,\!</math> is the number of groups of suspension data points
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in <math>{{i}^{th}}\,\!</math> group of suspension data points
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
:*<math>FI\,\!</math> is the number of interval failure data groups
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in <math>{{i}^{th}}\,\!</math> group of data intervals
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval


• <math>{{N}_{i}}</math> is the number of times-to-failure in the  <math>{{i}^{th}}</math> time-to-failure data group
For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.\,\!</math>


<math>\mu </math> is the Gumbel shape parameter (unknown a priori, the first of two parameters to be found)
The solution of the maximum log-likelihood function is found by solving for (<math>\widehat{\mu },\widehat{\sigma })\,\!</math> so that:


<math>\sigma </math> is the Gumbel scale parameter (unknown a priori, the second of two parameters to be found)
::<math>\tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial \sigma }=0.\,\!</math>


• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data


• <math>S</math> is the number of groups of suspension data points
::<math>\begin{align}
 
• <math>N_{i}^{\prime }</math> is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points
 
• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math>  suspension data group
 
• <math>FI</math> is the number of interval failure data groups
 
• <math>N_{i}^{\prime \prime }</math> is the number of intervals in <math>{{i}^{th}}</math> group of data intervals
 
• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval
 
• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval
 
For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.</math>
 
The solution of the maximum log-likelihood function is found by solving for (<math>\widehat{\mu },\widehat{\sigma })</math> so that:
 
<math>\tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial \sigma }=0.</math>
 
 
<math>\begin{align}
   \frac{\partial \Lambda }{\partial \mu }= & -\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}} \\  
   \frac{\partial \Lambda }{\partial \mu }= & -\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}} \\  
   & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}}{{{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}} \right)   
   & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}}{{{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}} \right)   
\end{align}</math>
\end{align}\,\!</math>






 
::<math>\begin{align}
<math>\begin{align}
   \frac{\partial \Lambda }{\partial \sigma }= & -\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{T}_{i}}-\mu }{{{\sigma }^{2}}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{T}_{i}}-\mu }{\sigma }{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}} \\  
   \frac{\partial \Lambda }{\partial \sigma }= & -\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{T}_{i}}-\mu }{{{\sigma }^{2}}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{T}_{i}}-\mu }{\sigma }{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}} \\  
   & -\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{T_{i}^{^{\prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }} \\  
   & -\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{T_{i}^{^{\prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }} \\  
   & \left( \frac{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}}{{{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}} \right)   
   & \left( \frac{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}}{{{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}} \right)   
\end{align}</math>
\end{align}\,\!</math>


===  The Gamma Log-Likelihood Functions and their Partials===
===  The Gamma Log-Likelihood Functions and their Partials===
This log-likelihood function is composed of three summation portions:
This log-likelihood function is composed of three summation portions:


<math>\begin{align}
::<math>\begin{align}
   \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{k(\ln ({{T}_{i}})-\mu )-{{e}^{{{e}^{\ln ({{T}_{i}})-\mu }}}}}}}{{{T}_{i}}\Gamma (k)} \right) \\  
   \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{k(\ln ({{T}_{i}})-\mu )-{{e}^{{{e}^{\ln ({{T}_{i}})-\mu }}}}}}}{{{T}_{i}}\Gamma (k)} \right) \\  
   & +\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln \left( 1-\Gamma \left( _{1}k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu )}} \right) \right) \\  
   & +\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln \left( 1-\Gamma \left( _{1}k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu )}} \right) \right) \\  
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right) \right)   
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right) \right)   
\end{align}</math>
\end{align}\,\!</math>


or:
or:


<math>\begin{align}
::<math>\begin{align}
   \Lambda = & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln ({{T}_{i}})\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln (\Gamma (k))+k\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}(\ln ({{T}_{i}})-\mu ) \\  
   \Lambda = & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln ({{T}_{i}})\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln (\Gamma (k))+k\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}(\ln ({{T}_{i}})-\mu ) \\  
   & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{e}^{\ln ({{T}_{i}})-\mu }} \\  
   & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{e}^{\ln ({{T}_{i}})-\mu }} \\  
   & +\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln \left( 1-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }} \right) \right) \\  
   & +\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln \left( 1-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }} \right) \right) \\  
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu )}} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu )}} \right) \right)   
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu )}} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu )}} \right) \right)   
\end{align}</math>
\end{align}\,\!</math>




where:
where:
<math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
:*<math>{{F}_{e}}\,\!</math> is the number of groups of times-to-failure data points
 
:*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group
<math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
:*<math>\mu \,\!</math> is the gamma shape parameter (unknown a priori, the first of two parameters to be found)
 
:*<math>k\,\!</math> is the gamma scale parameter (unknown a priori, the second of two parameters to be found)
<math>\mu </math> is the gamma shape parameter (unknown a priori, the first of two parameters to be found)
:*<math>{{T}_{i}}\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> group of time-to-failure data
 
:*<math>S\,\!</math> is the number of groups of suspension data points
<math>k</math> is the gamma scale parameter (unknown a priori, the second of two parameters to be found)
:*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in <math>{{i}^{th}}\,\!</math> group of suspension data points
 
:*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group
<math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
:*<math>FI\,\!</math> is the number of interval failure data groups
 
:*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in <math>{{i}^{th}}\,\!</math> group of data intervals
<math>S</math> is the number of groups of suspension data points
:*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval
 
:*and <math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval
• .. is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points
 
<math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group
 
<math>FI</math> is the number of interval failure data groups
 
<math>N_{i}^{\prime \prime }</math> is the number of intervals in <math>{{i}^{th}}</math> group of data intervals
 
<math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval
 
and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval


For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.</math>
For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.\,\!</math>


The solution of the maximum log-likelihood function is found by solving for (<math>\widehat{\mu },\widehat{\sigma })</math> so that <math>\tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial k}=0.</math>
The solution of the maximum log-likelihood function is found by solving for (<math>\widehat{\mu },\widehat{\sigma })\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial k}=0.\,\!</math>




<math>\begin{align}
::<math>\begin{align}
   \frac{\partial \Lambda }{\partial \mu }= & -k\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{e}^{\ln ({{T}_{i}})-\mu }} \\  
   \frac{\partial \Lambda }{\partial \mu }= & -k\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{e}^{\ln ({{T}_{i}})-\mu }} \\  
   & +\frac{1}{\Gamma (k)}\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{{{e}^{k\left( \ln (T_{i}^{^{\prime }})-\mu )-{{e}^{\ln (T_{i}^{^{\prime }})-\mu )}} \right)}}}{1-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }} \right)} \\  
   & +\frac{1}{\Gamma (k)}\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{{{e}^{k\left( \ln (T_{i}^{^{\prime }})-\mu )-{{e}^{\ln (T_{i}^{^{\prime }})-\mu )}} \right)}}}{1-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }} \right)} \\  
   & +\frac{1}{\Gamma (k)}\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\{\frac{{{e}^{k{{e}^{{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }}}}-{{e}^{{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }}}}}}}{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right)} \\  
   & +\frac{1}{\Gamma (k)}\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\{\frac{{{e}^{k{{e}^{{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }}}}-{{e}^{{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }}}}}}}{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right)} \\  
   & -\frac{{{e}^{k{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }}-{{e}^{{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }}}}}}}{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right)}\}   
   & -\frac{{{e}^{k{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }}-{{e}^{{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }}}}}}}{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right)}\}   
\end{align}</math>
\end{align}\,\!</math>








<math>\begin{align}
::<math>\begin{align}
   \frac{\partial \Lambda }{\partial k}= & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}(\ln ({{T}_{i}})-\mu )-\frac{{{\Gamma }^{^{\prime }}}(k)\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\mathop{}_{}^{}}}\,}}\,{{N}_{i}}}{\Gamma (k)} \\  
   \frac{\partial \Lambda }{\partial k}= & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}(\ln ({{T}_{i}})-\mu )-\frac{{{\Gamma }^{^{\prime }}}(k)\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\mathop{}_{}^{}}}\,}}\,{{N}_{i}}}{\Gamma (k)} \\  
   & -\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{\tfrac{\partial {{\Gamma }_{1}}(k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }})}{\partial k}}{1-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }} \right)} \\  
   & -\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{\tfrac{\partial {{\Gamma }_{1}}(k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }})}{\partial k}}{1-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }} \right)} \\  
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{\tfrac{\partial {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right)}{\partial k}-\tfrac{\partial {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)}{\partial k}}{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }}) \right)} \right)   
   & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{\tfrac{\partial {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right)}{\partial k}-\tfrac{\partial {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)}{\partial k}}{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }}) \right)} \right)   
\end{align}</math>
\end{align}\,\!</math>

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Chapter Appendix D: Appendix: Log-Likelihood Equations


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Chapter Appendix D  
Appendix: Log-Likelihood Equations  

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Available Software:
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More Resources:
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This appendix covers the log-likelihood functions and their associated partial derivatives for most of the distributions available in Weibull++. These distributions are discussed in more detail in the chapter for each distribution.

Weibull Log-Likelihood Functions and their Partials

The Two-Parameter Weibull

This log-likelihood function is composed of three summation portions:

[math]\displaystyle{ \begin{align} \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{\eta } \right)}^{\beta }}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{\eta } \right)}^{\beta }} \\ & \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}} \right] \end{align}\,\! }[/math]

where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ \beta \,\! }[/math] is the Weibull shape parameter (unknown a priori, the first of two parameters to be found)
  • [math]\displaystyle{ \eta \,\! }[/math] is the Weibull scale parameter (unknown a priori, the second of two parameters to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ FI\,\! }[/math] is the number of interval failure data groups
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

For the purposes of MLE, left censored data will be considered to be intervals with [math]\displaystyle{ T_{Li}^{\prime \prime }=0.\,\! }[/math]

The solution will be found by solving for a pair of parameters [math]\displaystyle{ \left( \widehat{\beta },\widehat{\eta } \right)\,\! }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \beta }=0\,\! }[/math] and [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \eta }=0.\,\! }[/math] It should be noted that other methods can also be used, such as direct maximization of the likelihood function, without having to compute the derivatives.


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}}{\eta } \right) \\ & -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{\eta } \right)-\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{\eta } \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{\eta } \right) \\ & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right){{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}+{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right){{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}} \end{align}\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \eta }= & \frac{-\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta }} \\ & +\frac{\beta }{\eta }\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{\eta } \right)}^{\beta }} \\ & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{\beta }{\eta } \right){{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-\left( \tfrac{\beta }{\eta } \right){{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{\eta } \right)}^{\beta }}}}} \end{align}\,\! }[/math]

The Three-Parameter Weibull

This log-likelihood function is again composed of three summation portions:

[math]\displaystyle{ \begin{align} \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }} \\ & \\ & +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}} \right] \end{align}\,\! }[/math]

where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ \beta \,\! }[/math] is the Weibull shape parameter (unknown a priori, the first of three parameters to be found)
  • [math]\displaystyle{ \eta \,\! }[/math] is the Weibull scale parameter (unknown a priori, the second of three parameters to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ \gamma \,\! }[/math] is the Weibull location parameter (unknown a priori, the third of three parameters to be found)
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ FI\,\! }[/math] is the number of interval data groups
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • and [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

The solution is found by solving for [math]\displaystyle{ \left( \widehat{\beta },\widehat{\eta },\widehat{\gamma } \right)\,\! }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \beta }=0,\,\! }[/math] [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \eta }=0,\,\! }[/math] and [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \gamma }=0.\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}-\gamma }{\eta } \right)-\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}-\gamma }{\eta } \right) \\ & -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right) \\ & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} \\ & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} \end{align}\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \eta }= & \frac{-\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{\eta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}+\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }}\left( \frac{\beta }{\eta } \right) \\ & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\tfrac{\beta }{\eta }{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} \\ & -\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\tfrac{\beta }{\eta }{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right){{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} \end{align}\,\! }[/math]



[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \gamma }= & \left( 1-\beta \right)\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,\left( \frac{{{N}_{i}}}{{{T}_{i}}-\gamma } \right)+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}-\gamma }{\eta } \right)}^{\beta }}\left( \frac{\beta }{{{T}_{i}}-\gamma } \right) \\ & +\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }-\gamma }{\eta } \right)}^{\beta }}\left( \frac{\beta }{T_{i}^{\prime }-\gamma } \right) \\ & +\underset{i=1}{\overset{FI}{\mathop{\sum }}}\,N_{i}^{\prime \prime }\frac{\tfrac{\beta }{T_{Li}^{\prime \prime }-\gamma }{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-\tfrac{\beta }{T_{Ri}^{\prime \prime }-\gamma }{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}-{{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }-\gamma }{\eta } \right)}^{\beta }}}}} \end{align}\,\! }[/math]


It should be pointed out that the solution to the three-parameter Weibull via MLE is not always stable and can collapse if [math]\displaystyle{ \beta \sim 1.\,\! }[/math] In estimating the true MLE of the three-parameter Weibull distribution, two difficulties arise. The first is a problem of non-regularity and the second is the parameter divergence problem, as discussed in Hirose [14].

Non-regularity occurs when [math]\displaystyle{ \beta \le 2.\,\! }[/math] In general, there are no MLE solutions in the region of [math]\displaystyle{ 0\lt \beta \lt 1.\,\! }[/math] When [math]\displaystyle{ 1\lt \beta \lt 2,\,\! }[/math] MLE solutions exist but are not asymptotically normal, as discussed in Hirose [14]. In the case of non-regularity, the solution is treated anomalously.

Weibull++ attempts to find a solution in all of the regions using a variety of methods, but the user should be forewarned that not all possible data can be addressed. Thus, some solutions using MLE for the three-parameter Weibull will fail when the algorithm has reached predefined limits or fails to converge. In these cases, the user can change to the non-true MLE approach (in Weibull++ Application Setup), where [math]\displaystyle{ \gamma \,\! }[/math] is estimated using non-linear regression. Once [math]\displaystyle{ \gamma \,\! }[/math] is obtained, the MLE estimates of [math]\displaystyle{ \widehat{\beta }\,\! }[/math] and [math]\displaystyle{ \widehat{\eta }\,\! }[/math] are computed using the transformation [math]\displaystyle{ T_{i}^{\prime }=({{T}_{i}}-\gamma ).\,\! }[/math]

Exponential Log-Likelihood Functions and their Partials

The One-Parameter Exponential

This log-likelihood function is composed of three summation portions:

[math]\displaystyle{ \ln (L)=\Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-\lambda T_{Li}^{\prime \prime }}}-{{e}^{-\lambda T_{Ri}^{\prime \prime }}} \right]\,\! }[/math]

where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ \lambda \,\! }[/math] is the failure rate parameter (unknown a priori, the only parameter to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ FI\,\! }[/math] is the number of interval data groups
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

The solution will be found by solving for a parameter [math]\displaystyle{ \widehat{\lambda }\,\! }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \lambda }=0.\,\! }[/math] Note that for [math]\displaystyle{ FI=0\,\! }[/math] there exists a closed form solution.

[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \lambda }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{1}{\lambda }-{{T}_{i}} \right)-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }T_{i}^{\prime } \\ & -\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\left[ \frac{T_{Li}^{\prime \prime }{{e}^{-\lambda T_{Li}^{\prime \prime }}}-T_{Ri}^{\prime \prime }{{e}^{-\lambda T_{Ri}^{\prime \prime }}}}{{{e}^{-\lambda T_{Li}^{\prime \prime }}}-{{e}^{-\lambda T_{Ri}^{\prime \prime }}}} \right] \end{align}\,\! }[/math]

The Two-Parameter Exponential

This log-likelihood function for the two-parameter exponential distribution is very similar to that of the one-parameter distribution and is composed of three summation portions:


[math]\displaystyle{ \begin{align} & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda \left( {{T}_{i}}-\gamma \right)}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda \left( T_{i}^{\prime }-\gamma \right) \\ & & \ \ +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ {{e}^{-\lambda \left( T_{Li}^{\prime \prime }-\gamma \right)}}-{{e}^{-\lambda \left( T_{Ri}^{\prime \prime }-\gamma \right)}} \right], \end{align}\,\! }[/math]

where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ \lambda \,\! }[/math] is the failure rate parameter (unknown a priori, the first of two parameters to be found)
  • [math]\displaystyle{ \gamma \,\! }[/math] is the location parameter (unknown a priori, the second of two parameters to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ FI\,\! }[/math] is the number of interval data groups
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

To find the two-parameter solution, look at the partial derivatives [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \lambda } }[/math] and [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \gamma} }[/math]:

[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \lambda }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left[ \frac{1}{\lambda }-\left( {{T}_{i}}-\gamma \right) \right] \\ & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\left( T_{i}^{\prime }-\gamma \right) \\ & -\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\left[ \frac{\left( T_{Li}^{\prime \prime }-\gamma \right){{e}^{-\lambda \left( T_{Li}^{\prime \prime }-{{\gamma }_{0}} \right)}}-\left( T_{Ri}^{\prime \prime }-\gamma \right){{e}^{-\lambda \left( T_{Ri}^{\prime \prime }-\gamma \right)}}}{{{e}^{-\lambda \left( T_{Li}^{\prime \prime }-\gamma \right)}}-{{e}^{-\lambda \left( T_{Ri}^{\prime \prime }-\gamma \right)}}} \right] \end{align}\,\! }[/math]

and

[math]\displaystyle{ \begin{align}\frac{\partial \Lambda }{\partial \gamma }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\lambda +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\lambda \,\!\end{align} }[/math].

From here we see that [math]\displaystyle{ \frac{\partial \Lambda }{\partial \gamma} }[/math] is a positive, constant function of [math]\displaystyle{ \gamma }[/math]. As alluded to in the chapter on the exponential distribution, this implies that the log-likelihood function [math]\displaystyle{ \Lambda }[/math] is, for fixed [math]\displaystyle{ \lambda }[/math], an increasing function of [math]\displaystyle{ \gamma }[/math]. Thus the MLE for [math]\displaystyle{ \gamma }[/math] is its largest possible value [math]\displaystyle{ T_1 }[/math]. Therefore, to find the full MLE solution [math]\displaystyle{ (\widehat{\lambda },\widehat{\gamma}) }[/math] for the two-parameter exponential distribution, one should set [math]\displaystyle{ \gamma }[/math] equal to the first failure time and then find (numerically) a [math]\displaystyle{ \lambda }[/math] such that [math]\displaystyle{ \tfrac{\partial \Lambda}{\partial \lambda} = 0 }[/math].

The 3D Plot utility in Weibull++ further illustrates this behavior of the log-likelihood function, as shown next:

Appendixc 127.gif

Normal Log-Likelihood Functions and their Partials

The complete normal likelihood function (without the constant) is composed of three summation portions:


[math]\displaystyle{ \begin{align} \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{\sigma }\phi \left( \frac{{{T}_{i}}-\mu }{\sigma } \right) \right] \\ & +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{^{\prime }}\ln \left[ 1-\Phi \left( \frac{T_{i}^{^{\prime }}-\mu }{\sigma } \right) \right] \\ & \text{ }+\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{^{\prime \prime }}\ln \left[ \Phi \left( \frac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma } \right)-\Phi \left( \frac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma } \right) \right] \end{align}\,\! }[/math]


where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ \mu \,\! }[/math] is the mean parameter (unknown a priori, the first of two parameters to be found)
  • [math]\displaystyle{ \sigma \,\! }[/math] is the standard deviation parameter (unknown a priori, the second of two parameters to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ {{F}_{i}}\,\! }[/math] is the number of interval data groups
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

The solution will be found by solving for a pair of parameters [math]\displaystyle{ \left( {{\mu }_{0}},{{\sigma }_{0}} \right)\,\! }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \mu }=0\,\! }[/math] and [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \sigma }=0.\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \mu }= & \frac{1}{{{\sigma }^{2}}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}({{T}_{i}}-\mu ) \\ & +\frac{1}{\sigma }\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)}{1-\Phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)} \\ & -\frac{1}{\sigma }\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}{\Phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\Phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)} \end{align}\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \sigma }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( {{T}_{i}}-\mu \right)}^{2}}}{{{\sigma }^{3}}}-\frac{1}{\sigma } \right) \\ & +\frac{1}{\sigma }\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)}{1-\Phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)} \\ & -\frac{1}{\sigma }\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}{\Phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\Phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)} \end{align}\,\! }[/math]

where:

[math]\displaystyle{ \phi \left( x \right)=\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{{{x}^{2}}}{2}}}\,\! }[/math]

and:

[math]\displaystyle{ \Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt\,\! }[/math]


Complete Data

Note that for the normal distribution, and in the case of complete data only (as was shown in Basic Statistical Background), there exists a closed-form solution for both of the parameters or:


[math]\displaystyle{ \widehat{\mu }=\widehat{{\bar{T}}}=\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{T}_{i}}\,\! }[/math]

and:

[math]\displaystyle{ \begin{align} \hat{\sigma }_{T}^{2}= & \frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{({{T}_{i}}-\bar{T})}^{2}} \\ {{{\hat{\sigma }}}_{T}}= & \sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{({{T}_{i}}-\bar{T})}^{2}}} \end{align}\,\! }[/math]

Lognormal Log-Likelihood Functions and their Partials

The general log-likelihood function (without the constant) for the lognormal distribution is composed of three summation portions:


[math]\displaystyle{ \begin{align} \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{T}_{i}} {{\sigma }_{{{T}'}}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right] \\ & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right] \\ & \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ \Phi \left( \frac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \frac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right] \end{align}\,\! }[/math]

where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ {\mu }'\,\! }[/math] is the mean of the natural logarithms of the times-to-failure (unknown a priori, the first of two parameters to be found)
  • [math]\displaystyle{ {{\sigma }_{{{T}'}}}\,\! }[/math] is the standard deviation of the natural logarithms of the times-to-failure (unknown a priori, the second of two parameters to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ FI\,\! }[/math] is the number of interval data groups
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

The solution will be found by solving for a pair of parameters [math]\displaystyle{ \left( {\mu }',{{\sigma }_{{{T}'}}} \right)\,\! }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial {\mu }'}=0\,\! }[/math] and [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0\,\! }[/math]:


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial {\mu }'}= & \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,N_{i}(\ln ({{T}_{i}})-{\mu }') \\ & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} \\ & \ \ -\underset{i=1}{\overset{FI}{\mathop \sum }}\,\frac{N_{i}^{\prime \prime }}{\sigma }\frac{\phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} \end{align}\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,N_{i}\left( \frac{{{\left( \ln ({{T}_{i}})-{\mu }' \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) \\ & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} \\ & -\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} \end{align}\,\! }[/math]

where:

[math]\displaystyle{ \phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{{{x}^{2}}}{2}}}\,\! }[/math]

and:

[math]\displaystyle{ \Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt\,\! }[/math]

Mixed Weibull Log-Likelihood Functions and their Partials

The log-likelihood function (without the constant) is composed of three summation portions:

[math]\displaystyle{ \begin{align} \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}\frac{{{\beta }_{k}}}{{{\eta }_{k}}}{{\left( \frac{{{T}_{i}}}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}-1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right] \\ & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}{{e}^{-{{\left( \tfrac{T_{i}^{\prime }}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right] \\ & \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}\frac{{{\beta }_{k}}}{{{\eta }_{k}}}{{\left( \frac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{{\eta }_{k}}} \right)}^{{{\beta }_{k}}-1}}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right] \end{align}\,\! }[/math]

where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ Q\,\! }[/math] is the number of subpopulations
  • [math]\displaystyle{ {{\rho }_{k}}\,\! }[/math] is the proportionality of the [math]\displaystyle{ {{k}^{th}}\,\! }[/math] subpopulation (unknown a priori, the first set of three sets of parameters to be found)
  • [math]\displaystyle{ {{\beta }_{k}}\,\! }[/math] is the Weibull shape parameter of the [math]\displaystyle{ {{k}^{th}}\,\! }[/math] subpopulation (unknown a priori, the second set of three sets of parameters to be found)
  • [math]\displaystyle{ {{\eta }_{k}}\,\! }[/math] is the Weibull scale parameter (unknown a priori, the third set of three sets of parameters to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ FI\,\! }[/math] is the number of groups of interval data points
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

The solution will be found by solving for a group of parameters:

[math]\displaystyle{ \left( \widehat{{{\rho }_{1,}}}\widehat{{{\beta }_{1}}},\widehat{{{\eta }_{1}}},\widehat{{{\rho }_{2,}}}\widehat{{{\beta }_{2}}},\widehat{{{\eta }_{2}}},...,\widehat{{{\rho }_{Q,}}}\widehat{{{\beta }_{Q}}},\widehat{{{\eta }_{Q}}} \right)\,\! }[/math]

so that:

[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial {{\rho }_{1}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{1}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{1}}}=0 \\ \frac{\partial \Lambda }{\partial {{\rho }_{2}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{2}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{2}}}=0 \\ \vdots \\ \frac{\partial \Lambda }{\partial {{\rho }_{Q-1}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{Q-1}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{Q-1}}}=0 \\ \frac{\partial \Lambda }{\partial {{\beta }_{Q}}}= & 0,\text{ and }\frac{\partial \Lambda }{\partial {{\eta }_{Q}}}=0 \end{align}\,\! }[/math]

Logistic Log-Likelihood Functions and their Partials

This log-likelihood function is composed of three summation portions:

[math]\displaystyle{ \begin{align} \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}{\sigma {{(1+{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}})}^{2}}} \right)-\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln (1+{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}) \\ & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( \frac{1}{1+{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}-\frac{1}{1+{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}} \right) \end{align}\,\! }[/math]

where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ \mu \,\! }[/math] is the logistic shape parameter (unknown a priori, the first of two parameters to be found)
  • [math]\displaystyle{ \eta \,\! }[/math] is the logistic scale parameter (unknown a priori, the second of two parameters to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ FI\,\! }[/math] is the number of interval failure data group
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

For the purposes of MLE, left censored data will be considered to be intervals with [math]\displaystyle{ T_{Li}^{\prime \prime }=0.\,\! }[/math]

The solution of the maximum log-likelihood function is found by solving for ([math]\displaystyle{ \widehat{\mu },\widehat{\sigma })\,\! }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial \sigma }=0.\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \mu }= & -\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}} \\ & -\frac{\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\mathop{}_{}^{}}}\,}}\,N_{i}^{^{\prime \prime }}}{\sigma }+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}+\frac{{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}} \right) \end{align}\,\! }[/math]



[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \sigma }= & -\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{T}_{i}}-\mu }{{{\sigma }^{2}}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{\tfrac{{{T}_{i}}-\mu }{\sigma }{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}} \\ & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}} \\ & \frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}(\frac{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}+\frac{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{1+{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}} \\ & -\frac{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}-\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}{{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}) \end{align}\,\! }[/math]

The Loglogistic Log-Likelihood Functions and their Partials

This log-likelihood function is composed of three summation portions:

[math]\displaystyle{ \begin{align} \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}}{\sigma t{{(1+{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}})}^{2}}} \right) \\ & -\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln (1+{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}) \\ & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( \frac{1}{1+{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}-\frac{1}{1+{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}} \right) \end{align}\,\! }[/math]

where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ \mu \,\! }[/math] is the loglogistic shape parameter (unknown a priori, the first of two parameters to be found)
  • [math]\displaystyle{ \sigma \,\! }[/math] is the loglogistic scale parameter (unknown a priori, the second of two parameters to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ FI\,\! }[/math] is the number of interval failure data groups,
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

For the purposes of MLE, left censored data will be considered to be intervals with [math]\displaystyle{ T_{Li}^{\prime \prime }=0.\,\! }[/math]

The solution of the maximum log-likelihood function is found by solving for ([math]\displaystyle{ \widehat{\mu },\widehat{\sigma })\,\! }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial \sigma }=0.\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \mu }= & -\frac{\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\mathop{}_{}^{}}}\,}}\,{{N}_{i}}}{\sigma }+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}} \\ & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}}-\frac{{{F}_{I}}}{\sigma } \\ & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}+\frac{{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}} \right) \end{align}\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \sigma }= & -\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{\ln ({{T}_{i}})-\mu }{{{\sigma }^{2}}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{2}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln ({{T}_{i}})-\mu }{\sigma }}}} \\ & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{i}^{^{\prime }})-\mu }{\sigma }}}} \\ & \frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}(\frac{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}+\frac{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{1+{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}} \\ & -\frac{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}-\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}{{{e}^{\tfrac{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}-{{e}^{\tfrac{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }{\sigma }}}}) \end{align}\,\! }[/math]

The Gumbel Log-Likelihood Functions and their Partials

This log-likelihood function is composed of three summation portions:

[math]\displaystyle{ \begin{align} \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }-{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}}}}{\sigma } \right) \\ & -\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln \left( {{e}^{-{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}}} \right) \\ & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}} \right) \end{align}\,\! }[/math]

or:

[math]\displaystyle{ \begin{align} \Lambda = & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( \frac{{{T}_{i}}-\mu }{\sigma }-{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}} \right)-\ln (\sigma )\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}} \\ & +\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}} \\ & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}} \right) \end{align}\,\! }[/math]

where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ \mu \,\! }[/math] is the Gumbel shape parameter (unknown a priori, the first of two parameters to be found)
  • [math]\displaystyle{ \sigma \,\! }[/math] is the Gumbel scale parameter (unknown a priori, the second of two parameters to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ FI\,\! }[/math] is the number of interval failure data groups
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

For the purposes of MLE, left censored data will be considered to be intervals with [math]\displaystyle{ T_{Li}^{\prime \prime }=0.\,\! }[/math]

The solution of the maximum log-likelihood function is found by solving for ([math]\displaystyle{ \widehat{\mu },\widehat{\sigma })\,\! }[/math] so that:

[math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial \sigma }=0.\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \mu }= & -\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}} \\ & +\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}}{{{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}} \right) \end{align}\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \sigma }= & -\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{T}_{i}}-\mu }{{{\sigma }^{2}}}-\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\frac{{{T}_{i}}-\mu }{\sigma }{{e}^{\tfrac{{{T}_{i}}-\mu }{\sigma }}} \\ & -\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{T_{i}^{^{\prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{i}^{^{\prime }}-\mu }{\sigma }}}+\frac{1}{\sigma }\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }} \\ & \left( \frac{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}}{{{e}^{-{{e}^{\tfrac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma }}}}}} \right) \end{align}\,\! }[/math]

The Gamma Log-Likelihood Functions and their Partials

This log-likelihood function is composed of three summation portions:

[math]\displaystyle{ \begin{align} \ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{e}^{k(\ln ({{T}_{i}})-\mu )-{{e}^{{{e}^{\ln ({{T}_{i}})-\mu }}}}}}}{{{T}_{i}}\Gamma (k)} \right) \\ & +\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln \left( 1-\Gamma \left( _{1}k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu )}} \right) \right) \\ & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right) \right) \end{align}\,\! }[/math]

or:

[math]\displaystyle{ \begin{align} \Lambda = & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln ({{T}_{i}})\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln (\Gamma (k))+k\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}(\ln ({{T}_{i}})-\mu ) \\ & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{e}^{\ln ({{T}_{i}})-\mu }} \\ & +\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\ln \left( 1-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }} \right) \right) \\ & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu )}} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu )}} \right) \right) \end{align}\,\! }[/math]


where:

  • [math]\displaystyle{ {{F}_{e}}\,\! }[/math] is the number of groups of times-to-failure data points
  • [math]\displaystyle{ {{N}_{i}}\,\! }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] time-to-failure data group
  • [math]\displaystyle{ \mu \,\! }[/math] is the gamma shape parameter (unknown a priori, the first of two parameters to be found)
  • [math]\displaystyle{ k\,\! }[/math] is the gamma scale parameter (unknown a priori, the second of two parameters to be found)
  • [math]\displaystyle{ {{T}_{i}}\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of time-to-failure data
  • [math]\displaystyle{ S\,\! }[/math] is the number of groups of suspension data points
  • [math]\displaystyle{ N_{i}^{\prime }\,\! }[/math] is the number of suspensions in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of suspension data points
  • [math]\displaystyle{ T_{i}^{\prime }\,\! }[/math] is the time of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] suspension data group
  • [math]\displaystyle{ FI\,\! }[/math] is the number of interval failure data groups
  • [math]\displaystyle{ N_{i}^{\prime \prime }\,\! }[/math] is the number of intervals in [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group of data intervals
  • [math]\displaystyle{ T_{Li}^{\prime \prime }\,\! }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval
  • and [math]\displaystyle{ T_{Ri}^{\prime \prime }\,\! }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval

For the purposes of MLE, left censored data will be considered to be intervals with [math]\displaystyle{ T_{Li}^{\prime \prime }=0.\,\! }[/math]

The solution of the maximum log-likelihood function is found by solving for ([math]\displaystyle{ \widehat{\mu },\widehat{\sigma })\,\! }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial k}=0.\,\! }[/math]


[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \mu }= & -k\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{e}^{\ln ({{T}_{i}})-\mu }} \\ & +\frac{1}{\Gamma (k)}\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{{{e}^{k\left( \ln (T_{i}^{^{\prime }})-\mu )-{{e}^{\ln (T_{i}^{^{\prime }})-\mu )}} \right)}}}{1-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }} \right)} \\ & +\frac{1}{\Gamma (k)}\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\{\frac{{{e}^{k{{e}^{{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }}}}-{{e}^{{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }}}}}}}{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right)} \\ & -\frac{{{e}^{k{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }}-{{e}^{{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }}}}}}}{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right)}\} \end{align}\,\! }[/math]



[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial k}= & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}(\ln ({{T}_{i}})-\mu )-\frac{{{\Gamma }^{^{\prime }}}(k)\underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{\mathop{}_{}^{}}}\,}}\,{{N}_{i}}}{\Gamma (k)} \\ & -\underset{i=1}{\mathop{\overset{S}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime }}\frac{\tfrac{\partial {{\Gamma }_{1}}(k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }})}{\partial k}}{1-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }} \right)} \\ & +\underset{i=1}{\mathop{\overset{{{F}_{I}}}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{\tfrac{\partial {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }} \right)}{\partial k}-\tfrac{\partial {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)}{\partial k}}{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{R}_{i}}}^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{{{L}_{i}}}^{^{\prime \prime }})-\mu }}) \right)} \right) \end{align}\,\! }[/math]