Template:Lognormal distribution fisher matrix bounds: Difference between revisions

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

Latest revision as of 06:02, 13 August 2012

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