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, $${\widehat{\sigma}'}$$,  $$\ln $$  is treated as normally distributed, and the bounds are estimated from:


 * $$\begin{align}

& {{\sigma}_{U}}= & \cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var}}}}\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  $$$$  are estimated as follows:


 * $$\left( \begin{matrix}

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

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

Bounds on Reliability
The reliability of the lognormal distribution is:


 * $$\hat{R}(t;{{\hat{\mu }}^{'}},{{\hat{\sigma }}^{'}})=\int_{t'}^{\infty }{\frac{1}{{^{'}}\sqrt{2\pi }}}{{e}^{-\frac{1}{2}{{\left( \frac{x-{{{\hat{\mu }}}^{'}}}{{{{\hat{\sigma }}}^{'}}} \right)}^{2}}}}dx$$

where $$t'=\ln (t)$$. Let $$\hat{z}(x)=\frac{x-{{{\hat{\mu }}}^{'}}}$$, the above equation then becomes:


 * $$\hat{R}\left( \hat{z}(t') \right)=\int_{\hat{z}(t')}^{\infty }{\frac{1}{\sqrt{2\pi }}}{{e}^{-\frac{1}{2}{{z}^{2}}}}dz$$

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


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

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:


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

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 $${T}'({{\widehat{\mu }}^{\prime }},):$$


 * $$\begin{align}

& Var({{{\hat{t}}}^{\prime }})= & {{\left( \frac{\partial {t}'}{\partial {\mu }'} \right)}^{2}}Var({{\widehat{\mu }}^{\prime }})+{{\left( \frac{\partial {t}'}{\partial } \right)}^{2}}Var \\ & & +2\left( \frac{\partial {t}'}{\partial {\mu }'} \right)\left( \frac{\partial {t}'}{\partial } \right)Cov\left( {{\widehat{\mu }}^{\prime }}, \right) \\ & &  \\  & Var({{{\hat{t}}}^{\prime }})= & Var({{\widehat{\mu }}^{\prime }})+{{\widehat{z}}^{2}}Var+2\cdot \widehat{z}\cdot Cov\left( {{\widehat{\mu }}^{\prime }}, \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: