Template:Normal Distribution fisher matrix confidence bounds - Revision history

Lisa Hacker: Redirected page to The Normal Distribution

2012-08-03T09:05:20Z

Redirected page to The Normal Distribution

← Older revision		Revision as of 09:05, 3 August 2012
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	~~===Fisher Matrix Confidence Bounds===~~		#REDIRECT [[The_Normal_Distribution]]
	~~====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}}~~

Harry Guo at 18:23, 13 February 2012

2012-02-13T18:23:05Z

← Older revision		Revision as of 18:23, 13 February 2012
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	& {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (lower bound)}		& {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (lower bound)}
	\end{align}</math>		\end{align}</math>


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

Harry Guo: /* Bounds on Time */

2012-02-10T21:27:40Z

Bounds on Time

← Older revision		Revision as of 21:27, 10 February 2012
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	::<math>\hat{T}(\widehat{\mu },{{\widehat{\sigma }}})=\widehat{\mu }+z\cdot {{\widehat{\sigma }}}</math>		::<math>\hat{T}(\widehat{\mu },{{\widehat{\sigma }}})=\widehat{\mu }+z\cdot {{\widehat{\sigma }}}</math>

	:where:		where:

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

	:and:		and:

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

Harry Guo: /* Bounds on Reliability */

2012-02-10T21:27:13Z

Bounds on Reliability

← Older revision		Revision as of 21:27, 10 February 2012
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	where:		where:

	::<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>		::<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:		or:

Harry Guo: /* Bounds on Time */

2012-02-10T21:26:21Z

Bounds on Time

← Older revision		Revision as of 21:26, 10 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial T}{\partial {{\sigma }}} \right)}^{2}}Var({{\widehat{\sigma }}}) \\		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 T}{\partial {{\sigma }}} \right)Cov\left( \widehat{\mu },{{\widehat{\sigma }}} \right) \\		& +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)		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>		\end{align}</math>

Harry Guo: /* Bounds on Time */

2012-02-10T21:25:23Z

Bounds on Time

← Older revision		Revision as of 21:25, 10 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial T}{\partial {{\sigma }}} \right)}^{2}}Var({{\widehat{\sigma }}}) \\		Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial T}{\partial {{\sigma }}} \right)}^{2}}Var({{\widehat{\sigma }}}) \\
	& +2\left( \frac{\partial hat{T}}{\partial \mu } \right)\left( \frac{\partial T}{\partial {{\sigma }}} \right)Cov\left( \widehat{\mu },{{\widehat{\sigma }}} \right) \\		& +2\left( \frac{\partial \hat{T}}{\partial \mu } \right)\left( \frac{\partial 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)		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>		\end{align}</math>

Harry Guo: /* Bounds on Time */

2012-02-10T21:24:30Z

Bounds on Time

← Older revision		Revision as of 21:24, 10 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial T}{\partial {{\sigma }}} \right)}^{2}}Var({{\widehat{\sigma }}}) \\		Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial T}{\partial {{\sigma }}} \right)}^{2}}Var({{\widehat{\sigma }}}) \\
	& +2\left( \frac{\partial T}{\partial \mu } \right)\left( \frac{\partial T}{\partial {{\sigma }}} \right)Cov\left( \widehat{\mu },{{\widehat{\sigma }}} \right) \\		& +2\left( \frac{\partial hat{T}}{\partial \mu } \right)\left( \frac{\partial 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)		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>		\end{align}</math>

Harry Guo: /* Bounds on Time */

2012-02-10T21:23:37Z

Bounds on Time

← Older revision		Revision as of 21:23, 10 February 2012
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	====Bounds on Time====		====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:		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 }~~}_{T~~}})=\widehat{\mu }+z\cdot {{\widehat{\sigma }~~}_{T~~}}</math>		::<math>\hat{T}(\widehat{\mu },{{\widehat{\sigma }}})=\widehat{\mu }+z\cdot {{\widehat{\sigma }}}</math>

	:where:		:where:
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	:and:		:and:

	::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z~~(T)~~}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>		::<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 }~~}_{T~~}})</math> or:		The next step is to calculate the variance of <math>\hat{T}(\widehat{\mu },{{\widehat{\sigma }}})</math> or:


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

Harry Guo: /* Bounds on Reliability */

2012-02-10T21:19:30Z

Bounds on Reliability

← Older revision		Revision as of 21:19, 10 February 2012
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	::<math>\widehat{R}(T;\hat{\mu },{{\hat{\sigma }~~}_{T~~}})=\int_{T}^{\infty }\frac{1}{{{\widehat{\sigma }~~}_{T~~}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\widehat{\mu }}{{{\widehat{\sigma }~~}_{T~~}}} \right)}^{2}}}}dt</math>		::<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}~~(t;\hat{\mu },{{\hat{\sigma }}_{T}})~~=\tfrac{t-\widehat{\mu }}{{{\widehat{\sigma }~~}_{T}}},</math> then <math>\tfrac{dz}{dt}=\tfrac{1}{{{\widehat{\sigma }}_{T}}}.</math> For <math>t=T</math> , <math>\widehat{z}=\tfrac{T-\widehat{\mu }}{{{\widehat{\sigma }}_{T~~}}}</math> , ~~and for <math>t=\infty ,</math> <math>\widehat{z}=\infty .</math> The~~ above equation then becomes:		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>		::<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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	\end{align}</math>		\end{align}</math>

	:where:		where:

	::<math>Var(\widehat{z})={{\left( \frac{\partial z}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial z}{\partial {{\sigma ~~}_{T~~}}} \right)}^{2}}Var({{\widehat{\sigma }~~}_{T~~}})+2\left( \frac{\partial z}{\partial \mu } \right)\left( \frac{\partial z}{\partial {{\sigma ~~}_{T~~}}} \right)Cov\left( \widehat{\mu },{{\widehat{\sigma }~~}_{T~~}} \right)</math>		::<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>

	:or:		or:

	::<math>Var(\widehat{z})=\frac{1}{\widehat{\sigma ~~}_{T~~}^{2}}\left[ Var(\widehat{\mu })+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }~~}_{T~~}})+2\cdot \widehat{z}\cdot Cov\left( \widehat{\mu },{{\widehat{\sigma }~~}_{T~~}} \right) \right]</math>		::<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>

Harry Guo at 21:12, 10 February 2012

2012-02-10T21:12:41Z

← Older revision		Revision as of 21:12, 10 February 2012
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	<math>\Lambda </math> is the log-likelihood function of the normal distribution, described in		<math>\Lambda </math> is the log-likelihood function of the normal distribution, described in
	Chapter 3 and Appendix C.		Chapter [[Parameter Estimation]] and [[Appendix: Distribution Log-Likelihood Equations]].

	====Bounds on Reliability====		====Bounds on Reliability====