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:


 * $$\begin{align}

\ln (L)= & \Lambda =\underset{i=1}{\mathop{\overset{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln \left( \frac{{{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{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu }} \right) \right)  \end{align}$$


 * or:


 * $$\begin{align}

\Lambda = & \underset{i=1}{\mathop{\overset{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln ({{T}_{i}})\underset{i=1}{\mathop{\overset{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln (\Gamma (k))+k\underset{i=1}{\mathop{\overset{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}(\ln ({{T}_{i}})-\mu ) \\ & \underset{i=1}{\mathop{\overset{\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{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu )}} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu )}} \right) \right) \end{align}$$


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

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

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


 * $$\begin{align}

\frac{\partial \Lambda }{\partial \mu }= & -k\underset{i=1}{\mathop{\overset{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}+\underset{i=1}{\mathop{\overset{\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{1-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_{i}^{^{\prime }})-\mu }} \right)} \\ & +\frac{1}{\Gamma (k)}\underset{i=1}{\mathop{\overset{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\{\frac{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu }} \right)} \\ & -\frac{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu }} \right)}\} \end{align}$$


 * $$\begin{align}

\frac{\partial \Lambda }{\partial k}= & \underset{i=1}{\mathop{\overset{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}(\ln ({{T}_{i}})-\mu )-\frac{{{\Gamma }^{^{\prime }}}(k)\underset{i=1}{\mathop{\overset{\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{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\left( \frac{\tfrac{\partial {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu }} \right)}{\partial k}-\tfrac{\partial {{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu }} \right)}{\partial k}}{{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu }} \right)-{{\Gamma }_{1}}\left( k;{{e}^{\ln (T_^{^{\prime \prime }})-\mu }}) \right)} \right) \end{align}$$