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| ===Fisher Matrix Bounds===
| | #REDIRECT [[The_Lognormal_Distribution#Fisher_Matrix_Bounds]] |
| ====Bounds on the Parameters====
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| The lower and upper bounds on the mean, <math>{\mu }'</math> , are estimated from:
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| ::<math>\begin{align}
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| & \mu _{U}^{\prime }= & {{\widehat{\mu }}^{\prime }}+{{K}_{\alpha }}\sqrt{Var({{\widehat{\mu }}^{\prime }})}\text{ (upper bound),} \\
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| & \mu _{L}^{\prime }= & {{\widehat{\mu }}^{\prime }}-{{K}_{\alpha }}\sqrt{Var({{\widehat{\mu }}^{\prime }})}\text{ (lower bound)}\text{.}
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| \end{align}</math>
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| For the standard deviation, <math>{\widehat{\sigma}'}</math> , <math>\ln ({{\widehat{\sigma'}}})</math> is treated as normally distributed, and the bounds are estimated from:
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| ::<math>\begin{align}
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| & {{\sigma}_{U}}= & {{\widehat{\sigma'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma'}}})}}{{{\widehat{\sigma'}}}}}}\text{ (upper bound),} \\
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| & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma' }}})}}{{{\widehat{\sigma'}}}}}}}\text{ (lower bound),}
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| \end{align}</math>
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| where <math>{{K}_{\alpha }}</math> is defined by:
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| ::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>
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| If <math>\delta </math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds and <math>\alpha =1-\delta </math> for the one-sided bounds.
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| The variances and covariances of <math>{{\widehat{\mu }}^{\prime }}</math> and <math>{{\widehat{\sigma'}}}</math> are estimated as follows:
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| ::<math>\left( \begin{matrix}
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| \widehat{Var}\left( {{\widehat{\mu }}^{\prime }} \right) & \widehat{Cov}\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma'}}} \right) \\
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| \widehat{Cov}\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma'}}} \right) & \widehat{Var}\left( {{\widehat{\sigma'}}} \right) \\
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| \end{matrix} \right)=\left( \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{({\mu }')}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {\mu }'\partial {{\sigma'}}} \\
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| {} & {} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {\mu }'\partial {{\sigma'}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma'^{2}} \\
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| \end{matrix} \right)_{{\mu }'={{\widehat{\mu }}^{\prime }},{{\sigma'}}={{\widehat{\sigma'}}}}^{-1}</math>
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| where <math>\Lambda </math> is the log-likelihood function of the lognormal distribution.
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| ====Bounds on Reliability====
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| The reliability of the lognormal distribution is:
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| ::<math>\hat{R}(t;{{\hat{\mu }}^{'}},{{\hat{\sigma }}^{'}})=\int_{t'}^{\infty }{\frac{1}{{{{\hat{\sigma }}}^{'}}\sqrt{2\pi }}}{{e}^{-\frac{1}{2}{{\left( \frac{x-{{{\hat{\mu }}}^{'}}}{{{{\hat{\sigma }}}^{'}}} \right)}^{2}}}}dx</math>
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| where <math>t'=\ln (t)</math>. Let <math>\hat{z}(x)=\frac{x-{{{\hat{\mu }}}^{'}}}{{{\sigma }^{'}}}</math>, the above equation then becomes:
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| ::<math>\hat{R}\left( \hat{z}(t') \right)=\int_{\hat{z}(t')}^{\infty }{\frac{1}{\sqrt{2\pi }}}{{e}^{-\frac{1}{2}{{z}^{2}}}}dz</math>
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| The bounds on <math>z</math> are estimated from:
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| ::<math>\begin{align}
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| & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\
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| & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})}
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| \end{align}</math>
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| where:
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| ::<math>\begin{align}
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| & 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) \\
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| & +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)
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| \end{align}</math>
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| or:
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| ::<math>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]</math>
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| The upper and lower bounds on reliability are:
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| ::<math>\begin{align}
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| & {{R}_{U}}= & \int_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\
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| & {{R}_{L}}= & \int_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)}
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| \end{align}</math>
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| ====Bounds on Time====
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| 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:
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| ::<math>{t}'({{\widehat{\mu }}^{\prime }},{{\widehat{\sigma' }}})={{\widehat{\mu }}^{\prime }}+z\cdot {{\widehat{\sigma' }}}</math>
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| where:
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| ::<math>z={{\Phi }^{-1}}\left[ F({t}') \right]</math>
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| and:
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| ::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({t}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>
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| The next step is to calculate the variance of <math>{T}'({{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}}):</math>
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| ::<math>\begin{align}
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| & 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' }}}) \\
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| & & +2\left( \frac{\partial {t}'}{\partial {\mu }'} \right)\left( \frac{\partial {t}'}{\partial {{\sigma' }}} \right)Cov\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma' }}} \right) \\
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| & & \\
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| & 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)
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| \end{align}</math>
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| The upper and lower bounds are then found by:
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| ::<math>\begin{align}
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| & T_{U}^{\prime }= & \ln {{T}_{U}}={{{\hat{T}}}^{\prime }}+{{K}_{\alpha }}\sqrt{Var({{{\hat{T}}}^{\prime }})} \\
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| & T_{L}^{\prime }= & \ln {{T}_{L}}={{{\hat{T}}}^{\prime }}-{{K}_{\alpha }}\sqrt{Var({{{\hat{T}}}^{\prime }})}
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| \end{align}</math>
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| Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> we get:
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| ::<math>\begin{align}
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| & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (upper bound),} \\
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| & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (lower bound)}\text{.}
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| \end{align}</math>
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| '''Example 4:'''
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| {{Example: Lognormal Distribution MLE}}
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