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| ===Fisher Matrix Confidence Bounds===
| | #REDIRECT [[The_Normal_Distribution]] |
| ====Bounds on the Parameters====
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| The lower and upper bounds on the mean, <math>\widehat{\mu }</math> , are estimated from:
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| ::<math>\begin{align}
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| & {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound),} \\
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| & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)}\text{.}
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| \end{align}</math>
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| 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:
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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 }</math> and <math>{{\widehat{\sigma }}}</math> are estimated from the Fisher matrix, as follows:
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| ::<math>\left( \begin{matrix}
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| \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },{{\widehat{\sigma }}} \right) \\
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| \widehat{Cov}\left( \widehat{\mu },{{\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 },\sigma =\widehat{\sigma }}^{-1}</math>
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| <math>\Lambda </math> is the log-likelihood function of the normal distribution, described in
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| Chapter [[Parameter Estimation]] and [[Appendix: Distribution Log-Likelihood Equations]].
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| ====Bounds on Reliability====
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| The reliability of the normal distribution is:
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| ::<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>
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| Let <math>\widehat{z}=\tfrac{t-\widehat{\mu }}{{{\widehat{\sigma }}}}</math>, the above equation then becomes:
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| ::<math>\hat{R}(\widehat{z})=\int_{\widehat{z}(t)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{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>Var(\widehat{z})={{\left( \frac{\partial z}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\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 },{{\widehat{\sigma }}} \right)</math>
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| or:
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| ::<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>
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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 normal percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:
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| ::<math>\hat{T}(\widehat{\mu },{{\widehat{\sigma }}})=\widehat{\mu }+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}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>
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| The next step is to calculate the variance of <math>\hat{T}(\widehat{\mu },{{\widehat{\sigma }}})</math> or:
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| ::<math>\begin{align}
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| 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 }}}) \\
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| & +2\left( \frac{\partial \hat{T}}{\partial \mu } \right)\left( \frac{\partial \hat{T}}{\partial {{\sigma }}} \right)Cov\left( \widehat{\mu },{{\widehat{\sigma }}} \right) \\
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| Var(\hat{T})= & Var(\widehat{\mu })+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}})+2\cdot z\cdot Cov\left( \widehat{\mu },{{\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}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (upper bound)} \\
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| & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (lower bound)}
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| \end{align}</math>
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