Template:Lognormal distribution fisher matrix bounds

Bounds on the Parameters
The lower and upper bounds on the mean, $${\mu }'$$, are estimated from:


 * $$\begin{align}

& \mu _{U}^{\prime }= & {{\widehat{\mu }}^{\prime }}+{{K}_{\alpha }}\sqrt{Var({{\widehat{\mu }}^{\prime }})}\text{ (upper bound),} \\ & \mu _{L}^{\prime }= & {{\widehat{\mu }}^{\prime }}-{{K}_{\alpha }}\sqrt{Var({{\widehat{\mu }}^{\prime }})}\text{ (lower bound)}\text{.} \end{align}$$

For the standard deviation, $${{\widehat{\sigma }}_}$$,  $$\ln ({{\widehat{\sigma }}_})$$  is treated as normally distributed, and the bounds are estimated from:


 * $$\begin{align}

& {{\sigma }_{U}}= & {{\widehat{\sigma }}_}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_})}}}}\text{ (upper bound),} \\ & {{\sigma }_{L}}= & \frac\text{ (lower bound),} \end{align}$$

where $${{K}_{\alpha }}$$  is defined by:


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

If $$\delta $$  is the confidence level, then  $$\alpha =\tfrac{1-\delta }{2}$$  for the two-sided bounds and  $$\alpha =1-\delta $$  for the one-sided bounds.

The variances and covariances of $${{\widehat{\mu }}^{\prime }}$$  and  $${{\widehat{\sigma }}_}$$  are estimated as follows:


 * $$\left( \begin{matrix}

\widehat{Var}\left( {{\widehat{\mu }}^{\prime }} \right) & \widehat{Cov}\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}_} \right) \\ \widehat{Cov}\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}_} \right) & \widehat{Var}\left( {{\widehat{\sigma }}_} \right) \\ \end{matrix} \right)=\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{({\mu }')}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {\mu }'\partial {{\sigma }_}} \\ {} & {} \\   -\tfrac{{{\partial }^{2}}\Lambda }{\partial {\mu }'\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _^{2}}  \\ \end{matrix} \right)_{{\mu }'={{\widehat{\mu }}^{\prime }},{{\sigma }_}={{\widehat{\sigma }}_}}^{-1}$$

where $$\Lambda $$  is the log-likelihood function of the lognormal distribution.

Bounds on Reliability
The reliability of the lognormal distribution is:


 * $$\hat{R}({T}';{\mu }',{{\sigma }_})=\int_^{\infty }\frac{1}{{{\widehat{\sigma }}_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-{{\widehat{\mu }}^{\prime }}} \right)}^{2}}}}dt$$

Let $$\widehat{z}(t;{{\hat{\mu }}^{\prime }},{{\hat{\sigma }}_})=\tfrac{t-{{\widehat{\mu }}^{\prime }}},$$  then  $$\tfrac{d\widehat{z}}{dt}=\tfrac{1}.$$ For  $$t={T}'$$,  $$\widehat{z}=\tfrac{{T}'-{{\widehat{\mu }}^{\prime }}}$$ , and for  $$t=\infty ,$$   $$\widehat{z}=\infty .$$  The above equation then becomes:


 * $$\hat{R}(\widehat{z})=\int_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz$$

The bounds on $$z$$  are estimated from:


 * $$\begin{align}

& {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align}$$


 * where:


 * $$\begin{align}

& Var(\widehat{z})= & \left( \frac{\partial z}{\partial {\mu }'} \right)_^{2}Var({{\widehat{\mu }}^{\prime }})+\left( \frac{\partial z}{\partial {{\sigma }_}} \right)_^{2}Var({{\widehat{\sigma }}_}) \\ & & +2{{\left( \frac{\partial z}{\partial {\mu }'} \right)}_}{{\left( \frac{\partial z}{\partial {{\sigma }_}} \right)}_}Cov\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}_} \right) \end{align}$$


 * or:


 * $$Var(\widehat{z})=\frac{1}{\widehat{\sigma }_^{2}}\left[ Var({{\widehat{\mu }}^{\prime }})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_})+2\cdot \widehat{z}\cdot Cov\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}_} \right) \right]$$

The upper and lower bounds on reliability are:


 * $$\begin{align}

& {{R}_{U}}= & \int_^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \int_^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align}$$

Bounds on Time
The bounds around time for a given lognormal percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, as follows:


 * $${T}'({{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}_})={{\widehat{\mu }}^{\prime }}+z\cdot {{\widehat{\sigma }}_}$$


 * where:


 * $$z={{\Phi }^{-1}}\left[ F({T}') \right]$$


 * and:


 * $$\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz$$

The next step is to calculate the variance of $${T}'({{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}_}):$$


 * $$\begin{align}

& Var({{{\hat{T}}}^{\prime }})= & {{\left( \frac{\partial {T}'}{\partial {\mu }'} \right)}^{2}}Var({{\widehat{\mu }}^{\prime }})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_}} \right)}^{2}}Var({{\widehat{\sigma }}_}) \\ & & +2\left( \frac{\partial {T}'}{\partial {\mu }'} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_}} \right)Cov\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}_} \right) \\ & &  \\  & Var({{{\hat{T}}}^{\prime }})= & Var({{\widehat{\mu }}^{\prime }})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_})+2\cdot \widehat{z}\cdot Cov\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}_} \right) \end{align}$$

The upper and lower bounds are then found by:


 * $$\begin{align}

& T_{U}^{\prime }= & \ln {{T}_{U}}={{{\hat{T}}}^{\prime }}+{{K}_{\alpha }}\sqrt{Var({{{\hat{T}}}^{\prime }})} \\ & T_{L}^{\prime }= & \ln {{T}_{L}}={{{\hat{T}}}^{\prime }}-{{K}_{\alpha }}\sqrt{Var({{{\hat{T}}}^{\prime }})} \end{align}$$

Solving for $${{T}_{U}}$$  and  $${{T}_{L}}$$  we get:


 * $$\begin{align}

& {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (upper bound),} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (lower bound)}\text{.} \end{align}$$

Example 4
Using the data of Example 2 and assuming a lognormal distribution, estimate the parameters using the MLE method.

Solution to Example 4
In this example we have only complete data. Thus, the partials reduce to:


 * $$\begin{align}

& \frac{\partial \Lambda }{\partial {\mu }'}= & \frac{1}{\sigma _^{2}}\cdot \underset{i=1}{\overset{14}{\mathop \sum }}\,\ln ({{T}_{i}})-{\mu }'=0 \\ & \frac{\partial \Lambda }{\partial {{\sigma }_}}= & \underset{i=1}{\overset{14}{\mathop \sum }}\,\left( \frac{\ln ({{T}_{i}})-{\mu }'}{\sigma _^{3}}-\frac{1} \right)=0 \end{align}$$

Substituting the values of $${{T}_{i}}$$  and solving the above system simultaneously, we get:


 * $$\begin{align}

& {{{\hat{\sigma }}}_}= & 0.849 \\ & {{{\hat{\mu }}}^{\prime }}= & 3.516 \end{align}$$

Using Eqns. (mean) and (sdv) we get:


 * $$\overline{T}=\hat{\mu }=48.25\text{ hours}$$


 * and:


 * $${{\hat{\sigma }}_}=49.61\text{ hours}.$$

The variance/covariance matrix is given by:


 * $$\left[ \begin{matrix}

\widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0515 & {} & \widehat{Cov}\left( {{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma }}}_} \right)=0.0000 \\ {} & {} & {} \\   \widehat{Cov}\left( {{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma }}}_} \right)=0.0000 & {} & \widehat{Var}\left( {{{\hat{\sigma }}}_} \right)=0.0258  \\ \end{matrix} \right]$$

Note About Bias
See the discussion regarding bias with the normal distribution in Chapter 8 for information regarding parameter bias in the lognormal distribution.