Template:Loglogistic confidence bounds - Revision history

Lisa Hacker: Redirected page to The Loglogistic Distribution

2012-08-09T09:58:36Z

Redirected page to The Loglogistic Distribution

← Older revision		Revision as of 09:58, 9 August 2012
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	~~==Confidence Bounds==~~		#REDIRECT [[The_Loglogistic_Distribution]]
	The method used by the application in estimating the different types of confidence bounds for loglogistically distributed data is presented in this section. The complete derivations were presented in detail for a general function in Chapter [[~~Parameter Estimation~~]].

	~~===Bounds on the Parameters===~~
	~~The lower and upper bounds <math>{\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)}~~
	~~\end{align}</math>~~


	~~For paramter <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 }</math> and <math>\widehat{\sigma }</math> are estimated 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>~~


	~~where <math>\Lambda </math> is the log-likelihood function of the loglogistic distribution.~~

	~~===Bounds on Reliability===~~
	~~The reliability of the logistic distribution is:~~

	~~::<math>\widehat{R}=\frac{1}{1+\exp (\widehat{z})}</math>~~

	~~where:~~

	~~::<math>\widehat{z}=\frac{{t}'-\widehat{\mu }}{\widehat{\sigma }}</math>~~

	Here <math>0<t<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> , therefore <math>0<t'=\ln (t)<\infty </math> and <math>z</math> also is changing from <math>-\infty </math> till <math>+\infty </math> .

	~~The bounds on <math>z</math> are estimated from:~~

	~~::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})}</math>~~

	~~::<math>{{z}_{L}}=\widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})\text{ }}\text{ }</math>~~

	~~where:~~

	::<math>Var(\widehat{z})={{(\frac{\partial z}{\partial \mu })}^{2}}Var({{\widehat{\mu }}^{\prime }})+2(\frac{\partial z}{\partial \mu })(\frac{\partial z}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial z}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>

	~~or:~~

	~~::<math>Var(\widehat{z})=\frac{1}{{{\sigma }^{2}}}(Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma }))</math>~~

	~~The upper and lower bounds on reliability are:~~

	~~::<math>{{R}_{U}}=\frac{1}{1+{{e}^{{{z}_{L}}}}}\text{(Upper bound)}</math>~~

	~~::<math>{{R}_{L}}=\frac{1}{1+{{e}^{{{z}_{U}}}}}\text{(Lower bound)}</math>~~

	~~===Bounds on Time===~~
	~~The bounds around time for a given loglogistic percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, as follows:~~

	~~::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })={{e}^{\widehat{\mu }+\widehat{\sigma }z}}</math>~~

	~~where:~~

	~~::<math>z=\ln (1-R)-\ln (R)</math>~~

	~~or:~~

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

	~~Let:~~

	~~::<math>{u}=\ln (\hat{T})=\widehat{\mu }+\widehat{\sigma }z</math>~~

	~~then:~~

	~~::<math>{u}_{U}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>~~


	~~::<math>{u}_{L}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>~~

	~~where:~~

	::<math>Var(\widehat{u})={{(\frac{\partial u}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial u}{\partial \mu })(\frac{\partial u}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial u}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>

	~~or:~~

	~~::<math>Var(\widehat{u})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>~~

	~~The upper and lower bounds are then found by:~~

	~~::<math>{{T}_{U}}={{e}^{{{u}_{U}}}}\text{ (upper bound)}</math>~~

	~~::<math>{{T}_{L}}={{e}^{{{u}_{L}}}}\text{ (lower bound)}</math>~~

Harry Guo: /* Bounds on Time */

2012-02-20T17:58:59Z

Bounds on Time

← Older revision		Revision as of 17:58, 20 February 2012
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	Let:		Let:

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

	then:		then:

	::<math>\{u}_{U}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>		::<math>{u}_{U}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>


	::<math>\{u}_{L}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>		::<math>{u}_{L}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>

	where:		where:

Harry Guo: /* Bounds on Time */

2012-02-20T17:58:44Z

Bounds on Time

← Older revision		Revision as of 17:58, 20 February 2012
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	Let:		Let:

	::<math>\~~widehat~~{u}=\ln (\hat{T})=\widehat{\mu }+\widehat{\sigma }z</math>		::<math>\{u}=\ln (\hat{T})=\widehat{\mu }+\widehat{\sigma }z</math>

	then:		then:

	::<math>\~~widehat~~{u}_{U}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>		::<math>\{u}_{U}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>


	::<math>\~~widehat~~{u}_{L}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>		::<math>\{u}_{L}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>

	where:		where:

Harry Guo: /* Bounds on Time */

2012-02-20T17:57:33Z

Bounds on Time

← Older revision		Revision as of 17:57, 20 February 2012
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	then:		then:

	::<math>{widehat{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>		::<math>\widehat{u}_{U}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>


	::<math>{widehat{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>		::<math>\widehat{u}_{L}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>

	where:		where:

Harry Guo: /* Bounds on Time */

2012-02-20T17:57:04Z

Bounds on Time

← Older revision		Revision as of 17:57, 20 February 2012
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	or:		or:

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

	Let:		Let:

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

	then:		then:

	::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>		::<math>{widehat{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>


	::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>		::<math>{widehat{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>

	where:		where:

Harry Guo: /* Bounds on Reliability */

2012-02-20T17:53:59Z

Bounds on Reliability

← Older revision		Revision as of 17:53, 20 February 2012
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	::<math>\widehat{z}=\frac{{t}'-\widehat{\mu }}{\widehat{\sigma }}</math>		::<math>\widehat{z}=\frac{{t}'-\widehat{\mu }}{\widehat{\sigma }}</math>

	Here <math>0<t<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> , therefore <math>0<t'=\ln (t)<\infty </math> and <math>z</math> also is changing from <math>-\infty </math> till <math>+\infty </math> .The bounds on <math>z</math> are estimated from:		Here <math>0<t<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> , therefore <math>0<t'=\ln (t)<\infty </math> and <math>z</math> also is changing from <math>-\infty </math> till <math>+\infty </math> .

			The bounds on <math>z</math> are estimated from:

	::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})}</math>		::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})}</math>

Harry Guo: /* Confidence Bounds */

2012-02-20T17:53:08Z

Confidence Bounds

← Older revision		Revision as of 17:53, 20 February 2012
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	==Confidence Bounds==		==Confidence Bounds==
	The method used by the application in estimating the different types of confidence bounds for loglogistically distributed data is presented in this section. The complete derivations were presented in detail for a general function in Chapter 5.		The method used by the application in estimating the different types of confidence bounds for loglogistically distributed data is presented in this section. The complete derivations were presented in detail for a general function in Chapter [[Parameter Estimation]].

	===Bounds on the Parameters===		===Bounds on the Parameters===

Harry Guo at 17:47, 20 February 2012

2012-02-20T17:47:22Z

← Older revision		Revision as of 17:47, 20 February 2012
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	====Confidence Bounds====		==Confidence Bounds==
	The method used by the application in estimating the different types of confidence bounds for loglogistically distributed data is presented in this section. The complete derivations were presented in detail for a general function in Chapter 5.		The method used by the application in estimating the different types of confidence bounds for loglogistically distributed data is presented in this section. The complete derivations were presented in detail for a general function in Chapter 5.

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

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	where <math>\Lambda </math> is the log-likelihood function of the loglogistic distribution.		where <math>\Lambda </math> is the log-likelihood function of the loglogistic distribution.

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

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	::<math>{{R}_{L}}=\frac{1}{1+{{e}^{{{z}_{U}}}}}\text{(Lower bound)}</math>		::<math>{{R}_{L}}=\frac{1}{1+{{e}^{{{z}_{U}}}}}\text{(Lower bound)}</math>

	=====Bounds on Time=====		===Bounds on Time===
	The bounds around time for a given loglogistic percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, as follows:		The bounds around time for a given loglogistic percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, as follows:

Harry Guo: /* Bounds on Reliability */

2012-02-16T18:28:31Z

Bounds on Reliability

← Older revision		Revision as of 18:28, 16 February 2012
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	where:		where:

	::<math>\widehat{z}=\frac{{T}'-\widehat{\mu }}{\widehat{\sigma }}</math>		::<math>\widehat{z}=\frac{{t}'-\widehat{\mu }}{\widehat{\sigma }}</math>

	Here <math>0<t<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> , therefore <math>0<\ln (t)<\infty </math> and <math>z</math> also is changing from <math>-\infty </math> till <math>+\infty </math> .The bounds on <math>z</math> are estimated from:		Here <math>0<t<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> , therefore <math>0<t'=\ln (t)<\infty </math> and <math>z</math> also is changing from <math>-\infty </math> till <math>+\infty </math> .The bounds on <math>z</math> are estimated from:

	::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})}</math>		::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})}</math>

Harry Guo: /* Bounds on Time */

2012-02-15T02:05:53Z

Bounds on Time

← Older revision		Revision as of 02:05, 15 February 2012
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	::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })={{e}^{\widehat{\mu }+\widehat{\sigma }z}}</math>		::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })={{e}^{\widehat{\mu }+\widehat{\sigma }z}}</math>

			where:
	:where:

	::<math>z=\ln (1-R)-\ln (R)</math>		::<math>z=\ln (1-R)-\ln (R)</math>

			or:
	:or:

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

			Let:
	:Let:

	::<math>u=\ln (T)=\widehat{\mu }+\widehat{\sigma }z</math>		::<math>u=\ln (T)=\widehat{\mu }+\widehat{\sigma }z</math>

			then:
	:then:

	::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>		::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>




	::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>		::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>

			where:
	:where:


	::<math>Var(\widehat{u})={{(\frac{\partial u}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial u}{\partial \mu })(\frac{\partial u}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial u}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>		::<math>Var(\widehat{u})={{(\frac{\partial u}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial u}{\partial \mu })(\frac{\partial u}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial u}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>

			or:
	:or:

	::<math>Var(\widehat{u})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>		::<math>Var(\widehat{u})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>


	The upper and lower bounds are then found by:		The upper and lower bounds are then found by:

	::<math>{{T}_{U}}={{e}^{{{u}_{U}}}}\text{ (upper bound)}</math>		::<math>{{T}_{U}}={{e}^{{{u}_{U}}}}\text{ (upper bound)}</math>



	::<math>{{T}_{L}}={{e}^{{{u}_{L}}}}\text{ (lower bound)}</math>		::<math>{{T}_{L}}={{e}^{{{u}_{L}}}}\text{ (lower bound)}</math>