Template:Ald mle

MLE Parameter Estimation
The parameters of the lognormal distribution can also be estimated using Maximum Likelihood Estimation (MLE). This general log-likelihood function is:


 * $$\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]+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })] \end{align}$$

where:
 * $$z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-{\mu }'}{\sigma _{T}^{\prime }}$$


 * $$z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-{\mu }'}{\sigma _{T}^{\prime }}$$

and:
 * •	 $${{F}_{e}}$$ 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  $${{i}^{th}}$$  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.
 * •	 $$T_{Ri}^{\prime \prime }$$ 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  $$\tfrac{\partial \Lambda }{\partial {{\sigma }_}}=0,$$  where:
 * $$\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)}\overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\ & &  \\  & \frac{\partial \Lambda }{\partial {{\sigma }_}}= \underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\left( \frac{\sigma _^{3}}-\frac{1} \right)+\frac{1}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'} \right)}\overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \end{align}$$ and:
 * $$\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}$$


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