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:


 * $$\begin{align}

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


 * or


 * $$\begin{align}

\Lambda = & \underset{i=1}{\mathop{\overset{\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{\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{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{^{\prime \prime }}\ln \left( {{e}^{-{{e}^{\tfrac{T_^{^{\prime \prime }}-\mu }{\sigma }}}}}-{{e}^{-{{e}^{\tfrac{T_^{^{\prime \prime }}-\mu }{\sigma }}}}} \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 Gumbel shape parameter (unknown a priori, the first of two parameters to be found)
 * •	$$\sigma $$ is the Gumbel 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
 * •	$$N_{i}^{\prime }$$ 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:


 * $$\tfrac{\partial \Lambda }{\partial \mu }=0,\tfrac{\partial \Lambda }{\partial \sigma }=0.$$


 * $$\begin{align}

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


 * $$\begin{align}

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