|
|
| Line 1: |
Line 1: |
| | | #REDIRECT [[Appendix:_Log-Likelihood_Equations]] |
| === 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
| |
| ::• 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{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>
| |
| <br>
| |