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| | | #REDIRECT [[Appendix:_Log-Likelihood_Equations]] |
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| === The Gamma Log-Likelihood Functions and their Partials===
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| This log-likelihood function is composed of three summation portions:
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| ::<math>\begin{align}
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| \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) \\
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| & +\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) \\
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| & +\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)
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| \end{align}</math>
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| :or:
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| ::<math>\begin{align}
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| \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 ) \\
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| & \underset{i=1}{\mathop{\overset{{{F}_{e}}}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{e}^{\ln ({{T}_{i}})-\mu }} \\
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| & +\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) \\
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| & +\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)
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| \end{align}</math>
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| :where:
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| ::• <math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
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| ::• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
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| ::• <math>\mu </math> is the gamma shape parameter (unknown a priori, the first of two parameters to be found)
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| ::• <math>k</math> is the gamma scale parameter (unknown a priori, the second of two parameters to be found)
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| ::• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
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| ::• <math>S</math> is the number of groups of suspension data points
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| ::• .. is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points
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| ::• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group
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| ::• <math>FI</math> is the number of interval failure data groups
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| ::• <math>N_{i}^{\prime \prime }</math> is the number of intervals in <math>{{i}^{th}}</math> group of data intervals
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| ::• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval
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| ::• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval
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| For the purposes of MLE, left censored data will be considered to be intervals with <math>T_{Li}^{\prime \prime }=0.</math>
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| 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>
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| ::<math>\begin{align}
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| \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 }} \\
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| & +\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)} \\
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| & +\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)} \\
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| & -\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)}\}
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| \end{align}</math>
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| ::<math>\begin{align}
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| \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)} \\
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| & -\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)} \\
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| & +\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)
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| \end{align}</math>
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| <br>
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