View source for Template:Normal Distribution fisher matrix confidence bounds

===Fisher Matrix Confidence Bounds===
====Bounds on the Parameters====
The lower and upper bounds on the mean,  <math>\widehat{\mu }</math> , are estimated from: 

::<math>\begin{align}
  & {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound),} \\ 
 & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)}\text{.}  
\end{align}</math>


Since the standard deviation,  <math>{{\widehat{\sigma }}}</math> , must be positive,  <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 }</math>  and  <math>{{\widehat{\sigma }}}</math>  are estimated from the Fisher matrix, as follows:


::<math>\left( \begin{matrix}
   \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },{{\widehat{\sigma }}} \right)  \\
   \widehat{Cov}\left( \widehat{\mu },{{\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 },\sigma =\widehat{\sigma }}^{-1}</math>


<math>\Lambda </math>  is the log-likelihood function of the normal distribution, described in 
Chapter [[Parameter Estimation]] and [[Appendix: Distribution Log-Likelihood Equations]].

====Bounds on Reliability====
The reliability of the normal distribution is: 


::<math>\widehat{R}(t;\hat{\mu },{{\hat{\sigma }}})=\int_{t}^{\infty }\frac{1}{{{\widehat{\sigma }}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\widehat{\mu }}{{{\widehat{\sigma }}}} \right)}^{2}}}}dt</math>


Let <math>\widehat{z}=\tfrac{t-\widehat{\mu }}{{{\widehat{\sigma }}}}</math>, the above equation then becomes: 

::<math>\hat{R}(\widehat{z})=\int_{\widehat{z}(t)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{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>Var(\widehat{z})={{\left( \frac{\partial \hat{z}}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial \hat{z}}{\partial {{\sigma }}} \right)}^{2}}Var({{\widehat{\sigma }}})+2\left( \frac{\partial \hat{z}}{\partial \mu } \right)\left( \frac{\partial \hat{z}}{\partial {{\sigma }}} \right)Cov\left( \widehat{\mu },{{\widehat{\sigma }}} \right)</math>

or: 

::<math>Var(\widehat{z})=\frac{1}{\widehat{\sigma }^{2}}\left[ Var(\widehat{\mu })+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}})+2\cdot \widehat{z}\cdot Cov\left( \widehat{\mu },{{\widehat{\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 normal percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:


::<math>\hat{T}(\widehat{\mu },{{\widehat{\sigma }}})=\widehat{\mu }+z\cdot {{\widehat{\sigma }}}</math>

where:

::<math>z={{\Phi }^{-1}}\left[ F(T) \right]</math>

and:

::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>


The next step is to calculate the variance of  <math>\hat{T}(\widehat{\mu },{{\widehat{\sigma }}})</math>  or:


::<math>\begin{align}
   Var(\hat{T})= & {{\left( \frac{\partial \hat{T}}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial \hat{T}}{\partial {{\sigma }}} \right)}^{2}}Var({{\widehat{\sigma }}}) \\ 
   & +2\left( \frac{\partial \hat{T}}{\partial \mu } \right)\left( \frac{\partial \hat{T}}{\partial {{\sigma }}} \right)Cov\left( \widehat{\mu },{{\widehat{\sigma }}} \right) \\ 
  Var(\hat{T})= & Var(\widehat{\mu })+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}})+2\cdot z\cdot Cov\left( \widehat{\mu },{{\widehat{\sigma }}} \right)  
\end{align}</math>


The upper and lower bounds are then found by:


::<math>\begin{align}
  & {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (upper bound)} \\ 
 & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (lower bound)}  
\end{align}</math>


'''Example 4:'''
{{Example: Normal Distribution MLE}}