Template:Lognormal distribution fisher matrix bounds - Revision history

Richard House: Redirected page to The Lognormal Distribution#Fisher Matrix Bounds

2012-08-13T06:02:18Z

Redirected page to The Lognormal Distribution#Fisher Matrix Bounds

← Older revision		Revision as of 06:02, 13 August 2012
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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 Time(Type 1)====~~

	~~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' }}})={{\widehat{\mu }}^{\prime }}+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({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 }}}):</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' }}} \right)}^{2}}Var({{\widehat{\sigma' }}}) \\
	~~& & +2\left( \frac{\partial {t}'}{\partial {\mu }'} \right)\left( \frac{\partial {t}'}{\partial {{\sigma' }}} \right)Cov\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma' }}} \right) \\~~
	~~& & \\~~
	& 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)
	~~\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>~~

	~~====Bounds on Reliability (Type 2)====~~
	~~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 {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) \\
	~~& +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)~~
	~~\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>~~


	~~'''Example 4:'''~~
	~~{{Example: Lognormal Distribution MLE}}~~

Harry Guo: /* Bounds on Reliability (Type II) */

2012-02-22T23:59:56Z

Bounds on Reliability (Type II)

← Older revision		Revision as of 23:59, 22 February 2012
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	\end{align}</math>		\end{align}</math>

	====Bounds on Reliability (Type II)====		====Bounds on Reliability (Type 2)====
	The reliability of the lognormal distribution is:		The reliability of the lognormal distribution is:

Harry Guo: /* Bounds on Time(Type I) */

2012-02-22T23:59:39Z

Bounds on Time(Type I)

← Older revision		Revision as of 23:59, 22 February 2012
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	where <math>\Lambda </math> is the log-likelihood function of the lognormal distribution.		where <math>\Lambda </math> is the log-likelihood function of the lognormal distribution.

	====Bounds on Time(Type I)====		====Bounds on Time(Type 1)====

	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:		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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	& {{t}_{L}}= & {{e}^{t_{L}^{\prime }}}\text{ (lower bound)}\text{.}		& {{t}_{L}}= & {{e}^{t_{L}^{\prime }}}\text{ (lower bound)}\text{.}
	\end{align}</math>		\end{align}</math>


	====Bounds on Reliability (Type II)====		====Bounds on Reliability (Type II)====

Harry Guo: /* Fisher Matrix Bounds */

2012-02-22T23:59:13Z

Fisher Matrix Bounds

@@ Line 40: / Line 40: @@
 where  <math>\Lambda </math>  is the log-likelihood function of the lognormal distribution.
-====Bounds on Reliability====
+====Bounds on Time(Type I)====
 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:
@@ Line 113: / Line 79: @@
    & {{t}_{U}}= & {{e}^{t_{U}^{\prime }}}\text{ (upper bound),} \\
   & {{t}_{L}}= & {{e}^{t_{L}^{\prime }}}\text{ (lower bound)}\text{.}
 \end{align}</math>

Harry Guo: /* Bounds on Time */

2012-02-13T22:54:20Z

Bounds on Time

← Older revision		Revision as of 22:54, 13 February 2012
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	& {{t}_{L}}= & {{e}^{t_{L}^{\prime }}}\text{ (lower bound)}\text{.}		& {{t}_{L}}= & {{e}^{t_{L}^{\prime }}}\text{ (lower bound)}\text{.}
	\end{align}</math>		\end{align}</math>


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

Harry Guo: /* Bounds on Time */

2012-02-13T22:54:00Z

Bounds on Time

← Older revision		Revision as of 22:54, 13 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& T_{U}^{\prime }= & \ln {{T}_{U}}={{{\hat{T}}}^{\prime }}+{{K}_{\alpha }}\sqrt{Var({{{\hat{T}}}^{\prime }})} \\		& 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 }})}		& t_{L}^{\prime }= & \ln {{t}_{L}}={{{\hat{t}}}^{\prime }}-{{K}_{\alpha }}\sqrt{Var({{{\hat{t}}}^{\prime }})}
	\end{align}</math>		\end{align}</math>


	Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> we get:		Solving for <math>{{t}_{U}}</math> and <math>{{t}_{L}}</math> we get:

	::<math>\begin{align}		::<math>\begin{align}
	& {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (upper bound),} \\		& {{t}_{U}}= & {{e}^{t_{U}^{\prime }}}\text{ (upper bound),} \\
	& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (lower bound)}\text{.}		& {{t}_{L}}= & {{e}^{t_{L}^{\prime }}}\text{ (lower bound)}\text{.}
	\end{align}</math>		\end{align}</math>

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

Harry Guo: /* Bounds on Time */

2012-02-13T22:52:42Z

Bounds on Time

← Older revision		Revision as of 22:52, 13 February 2012
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	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({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>		::<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>		The next step is to calculate the variance of <math>{T}'({{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}}):</math>

	::<math>\begin{align}		::<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}~~'}}}) \\		& 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' }}}) \\
	& & +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) \\		& & +2\left( \frac{\partial {t}'}{\partial {\mu }'} \right)\left( \frac{\partial {t}'}{\partial {{\sigma' }}} \right)Cov\left( {{\widehat{\mu }}^{\prime }},{{\widehat{\sigma' }}} \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)		& 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)
	\end{align}</math>		\end{align}</math>

Harry Guo: /* Bounds on Time */

2012-02-13T22:50:15Z

Bounds on Time

← Older revision		Revision as of 22:50, 13 February 2012
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	::<math>{T}'({{\widehat{\mu }}^{\prime }},{{\widehat{\sigma ~~}}_{{{T}~~'}}})={{\widehat{\mu }}^{\prime }}+z\cdot {{\widehat{\sigma ~~}}_{{{T}~~'}}}</math>		::<math>{t}'({{\widehat{\mu }}^{\prime }},{{\widehat{\sigma' }}})={{\widehat{\mu }}^{\prime }}+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({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>		::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

Harry Guo: /* Bounds on Reliability */

2012-02-13T22:48:48Z

Bounds on Reliability

← Older revision		Revision as of 22:48, 13 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& 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) \\		& 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) \\
	& +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)		& +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)
	\end{align}</math>		\end{align}</math>
	or:		or:

Harry Guo: /* Bounds on Reliability */

2012-02-13T22:48:02Z

Bounds on Reliability

← Older revision		Revision as of 22:48, 13 February 2012
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	::<math>\begin{align}		::<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) \\		& 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) \\
	& +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)		& +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)
	\end{align}</math>		\end{align}</math>
	or:		or: