Appendix: Log-Likelihood Equations

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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]\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]{{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 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]{{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

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} \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]\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]\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]{{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

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 \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]\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, as discussed in Hirose [14].

Non-regularity occurs when [math]\beta \le 2.\,\![/math] In general, there are no MLE solutions in the region of [math]0\lt \beta \lt 1.\,\![/math] When [math]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]\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

The One-Parameter Exponential

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]

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]\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]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

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

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]\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]{{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

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] \\ & -\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]\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]

Examination the derivative for [math]\gamma\,\![/math] will reveal that:

[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 above equation will be equal to zero only if either:

[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:

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]\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]{{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

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

and:

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

and:

[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}}= & \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]\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]{{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 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]{{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

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

and:

[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

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

[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] \\ & \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]{{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]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]{{\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]{{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 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
  • [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:

[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:

[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 }_{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]\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]{{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 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]{{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 group
  • [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

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{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]\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]\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]{{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 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

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{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]\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]\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]\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]{{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

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{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]\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]\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]\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]{{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 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]{{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
  • 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 k}=0.\,\![/math]


[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{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]\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]