Lognormal Log-Likelihood Functions and their Partials

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:


 * $$\begin{align}

\ln (L)= & \Lambda =\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-{\mu }'} \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 }'} \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 }'} \right)-\Phi \left( \frac{\ln \left( T_{Li}^{\prime \prime } \right)-{\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 mean of the natural logarithms of the times-to-failure (unknown a priori, the first of two parameters to be found)
 * •	$${{\sigma }_}$$ is the standard deviation of the natural logarithms of the times-to-failure (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 the $${{i}^{th}}$$ group of suspension data points
 * •	$$T_{i}^{\prime }$$ is the time of the $${{i}^{th}}$$ suspension data group
 * •	$$FI$$ is the number of interval data groups
 * •	$$N_{i}^{\prime \prime }$$ is the number of intervals in the $${{i}^{th}}$$ 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

The solution will be found by solving for a pair of parameters $$\left( {\mu }',{{\sigma }_} \right)$$ so that $$\tfrac{\partial \Lambda }{\partial {\mu }'}=0$$ and $$\tfrac{\partial \Lambda }{\partial {{\sigma }_}}=0$$:


 * $$\begin{align}

\frac{\partial \Lambda }{\partial {\mu }'}= & \frac{1}{\sigma _^{2}}\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})-{\mu }') \\ & +\frac{1}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'} \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 }'} \right)-\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'} \right)} \end{align}$$ $$$$


 * where:


 * $$\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}$$


 * and:


 * $$\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{2}}}dt$$