Template:Logistic confidence bounds - Revision history

Lisa Hacker: Redirected page to The Logistic Distribution

2012-08-03T08:58:02Z

Redirected page to The Logistic Distribution

← Older revision		Revision as of 08:58, 3 August 2012
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	~~==Confidence Bounds==~~		#REDIRECT [[The_Logistic_Distribution]]
	In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. The complete derivations were presented in detail (for a general function) in Chapter [[~~Confidence Bounds~~]].

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

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

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

	~~The lower and upper bounds on the scale parameter <math>\widehat{\sigma }</math> are estimated from:~~

	~~::<math>{{\sigma }_{U}}=\widehat{\sigma }{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })\text{ }}}{\widehat{\sigma }}}}(\text{upper bound})</math>~~


	~~::<math>{{\sigma }_{L}}=\widehat{\sigma }{{e}^{\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })\text{ }}}{\widehat{\sigma }}}}\text{ (lower bound)}</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 logistic distribution is:~~

	~~::<math>\widehat{R}=\frac{1}{1+{{e}^{\widehat{z}}}}</math>~~

	~~where:~~

	~~::<math>\widehat{z}=\frac{t-\widehat{\mu }}{\widehat{\sigma }}</math>~~


	Here <math>-\infty <t<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> . Therefore, <math>z</math> also is changing from <math>-\infty </math> to <math>+\infty </math> . Then the bounds on <math>z</math> are estimated from:

	~~::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})\text{ }}</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 })+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 logistic percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:~~

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

	~~where,~~

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

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

	~~or:~~

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

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

Harry Guo: /* Bounds on Time */

2012-02-14T23:35:06Z

Bounds on Time

← Older revision		Revision as of 23:35, 14 February 2012
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	::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>		::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>

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

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

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

Harry Guo: /* Bounds on Time */

2012-02-14T23:34:42Z

Bounds on Time

← Older revision		Revision as of 23:34, 14 February 2012
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	::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>		::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>

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

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

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

Harry Guo: /* Bounds on Time */

2012-02-14T23:33:46Z

Bounds on Time

← Older revision		Revision as of 23:33, 14 February 2012
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	::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>		::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>

	:where:		where:

	::<math>z=\ln (1-R)-\ln (R)</math>		::<math>z=\ln (1-R)-\ln (R)</math>
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	::<math>Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>		::<math>Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>

	:or:		or:

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

Harry Guo: /* Bounds on Reliability */

2012-02-14T23:33:07Z

Bounds on Reliability

← Older revision		Revision as of 23:33, 14 February 2012
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	::<math>\widehat{R}=\frac{1}{1+{{e}^{\widehat{z}}}}</math>		::<math>\widehat{R}=\frac{1}{1+{{e}^{\widehat{z}}}}</math>

	:where:		where:

	::<math>\widehat{z}=\frac{T-\widehat{\mu }}{\widehat{\sigma }}</math>		::<math>\widehat{z}=\frac{t-\widehat{\mu }}{\widehat{\sigma }}</math>


	Here <math>-\infty <T<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> . Therefore, <math>z</math> also is changing from <math>-\infty </math> to <math>+\infty </math> . Then the bounds on <math>z</math> are estimated from:		Here <math>-\infty <t<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> . Therefore, <math>z</math> also is changing from <math>-\infty </math> to <math>+\infty </math> . Then the bounds on <math>z</math> are estimated from:

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

	:where:		where:

	::<math>Var(\widehat{z})={{(\frac{\partial z}{\partial \mu })}^{2}}Var(\widehat{\mu })+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>		::<math>Var(\widehat{z})={{(\frac{\partial z}{\partial \mu })}^{2}}Var(\widehat{\mu })+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:		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>		::<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>

Harry Guo: /* Confidence Bounds */

2012-02-14T23:32:05Z

Confidence Bounds

← Older revision		Revision as of 23:32, 14 February 2012
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	==Confidence Bounds==		==Confidence Bounds==
	In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. The complete derivations were presented in detail (for a general function) in Chapter 5.		In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. The complete derivations were presented in detail (for a general function) in Chapter [[Confidence Bounds]].

	===Bounds on the Parameters===		===Bounds on the Parameters===
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	::<math>{{\mu }_{U}}=\widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })\text{ }}\text{ (upper bound)}</math>		::<math>{{\mu }_{U}}=\widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })\text{ }}\text{ (upper bound)}</math>

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

	The lower and upper bounds on the scale parameter <math>\widehat{\sigma }</math> are estimated from:		The lower and upper bounds on the scale parameter <math>\widehat{\sigma }</math> are estimated from:
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	<math>\Lambda </math> is the log-likelihood function of the normal distribution, described in Chapter 3 and Appendix C.		<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===		===Bounds on Reliability===

Harry Guo: /* Confidence Bounds */

2012-02-14T22:49:55Z

Confidence Bounds

← Older revision		Revision as of 22:49, 14 February 2012
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	::<math>{{T}_{L}}=\widehat{T}-{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{lower bound})</math>		::<math>{{T}_{L}}=\widehat{T}-{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{lower bound})</math>

	~~{{logistic distribution example}}~~

Harry Guo at 22:40, 14 February 2012

2012-02-14T22:40:17Z

← Older revision		Revision as of 22:40, 14 February 2012
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	====Confidence Bounds====		==Confidence Bounds==
	In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. The complete derivations were presented in detail (for a general function) in Chapter 5.		In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. 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 on the location parameter <math>\widehat{\mu }</math> are estimated from		The lower and upper bounds on the location parameter <math>\widehat{\mu }</math> are estimated from
	:		:
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	<math>\Lambda </math> is the log-likelihood function of the normal distribution, described in Chapter 3 and Appendix C.		<math>\Lambda </math> is the log-likelihood function of the normal distribution, described in Chapter 3 and Appendix C.

	=====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 logistic percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:		The bounds around time for a given logistic percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:

Nicolette Young: /* A Logistic Distribution Example */

2012-01-19T22:30:51Z

A Logistic Distribution Example

← Older revision		Revision as of 22:30, 19 January 2012
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	::<math>{{T}_{L}}=\widehat{T}-{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{lower bound})</math>		::<math>{{T}_{L}}=\widehat{T}-{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{lower bound})</math>

	~~=====A Logistic Distribution Example=====~~		{{logistic distribution example}}
	~~The lifetime of a mechanical valve is known to follow a~~ logistic distribution~~. Ten units were tested for 28 months and the following months-to-failure data was collected.~~

	~~<center><math>\overset{{}}{\mathop{\text{Table 10}\text{.1 - Times-to-Failure Data with Suspensions}}}\,</math></center>~~
	~~<center><math>\begin{matrix}~~
	~~\text{Data Point Index} & \text{State F or S} & \text{State End Time} \\~~
	~~\text{1} & \text{F} & \text{8} \\~~
	~~\text{2} & \text{F} & \text{10} \\~~
	~~\text{3} & \text{F} & \text{15} \\~~
	~~\text{4} & \text{F} & \text{17} \\~~
	~~\text{5} & \text{F} & \text{19} \\~~
	~~\text{6} & \text{F~~} ~~& \text{26} \\~~
	~~\text{7} & \text{F} & \text{27} \\~~
	~~\text{8} & \text{S} & \text{28} \\~~
	~~\text{9} & \text{S} & \text{28} \\~~
	~~\text{10} & \text{S} & \text{28} \\~~
	~~\end{matrix}</math></center>~~

	~~:• Determine the valve's design life if specifications call for a reliability goal of 0.90.~~
	~~:• The valve is to be used in a pumping device that requires 1 month of continuous operation. What is the probability of the pump failing due to the valve?~~

	~~This data set can be entered into Weibull++ as follows:~~

	~~[[Image:sof-folio.png\|thumb\|center\|400px\| ]]~~

	~~The computed parameters for maximum likelihood are:~~

	~~::<math>\begin{align}~~
	~~& \widehat{\mu }= & 22.34 \\~~
	~~& \hat{\sigma }= & 6.15~~
	~~\end{align~~}~~</math>~~

	~~:• The valve's design life, along with 90% two sided confidence bounds, can be obtained using the QCP as follows:~~

	~~[[Image:ldaLD10.6.gif\|thumb\|center\|300px\| ]]~~

	~~:• The probability, along with 90% two sided confidence bounds, that the pump fails due to a valve failure during the first month is obtained as follows:~~


	~~[[Image:ldaLD10.7.gif\|thumb\|center\|300px\| ]]~~

Nicolette Young: Created page with '====Confidence Bounds==== In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. …'

2012-01-04T21:59:46Z

Created page with '====Confidence Bounds==== In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. …'

New page

====Confidence Bounds====
In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. The complete derivations were presented in detail (for a general function) in Chapter 5.

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

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

<math>{{\mu }_{L}}=\widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })\text{ }}\text{ (lower bound)}</math>

The lower and upper bounds on the scale parameter <math>\widehat{\sigma }</math> are estimated from:

::<math>{{\sigma }_{U}}=\widehat{\sigma }{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })\text{ }}}{\widehat{\sigma }}}}(\text{upper bound})</math>

::<math>{{\sigma }_{L}}=\widehat{\sigma }{{e}^{\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })\text{ }}}{\widehat{\sigma }}}}\text{ (lower bound)}</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 3 and Appendix C.

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

::<math>\widehat{R}=\frac{1}{1+{{e}^{\widehat{z}}}}</math>

:where:

::<math>\widehat{z}=\frac{T-\widehat{\mu }}{\widehat{\sigma }}</math>

Here <math>-\infty <T<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> . Therefore, <math>z</math> also is changing from <math>-\infty </math> to <math>+\infty </math> . Then the bounds on <math>z</math> are estimated from:

::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})\text{ }}</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 })+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 logistic percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:

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

:where:

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

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

:or:

::<math>Var(\widehat{T})=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}}=\widehat{T}+{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{upper bound})</math>

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

=====A Logistic Distribution Example=====
The lifetime of a mechanical valve is known to follow a logistic distribution. Ten units were tested for 28 months and the following months-to-failure data was collected.

<center><math>\overset{{}}{\mathop{\text{Table 10}\text{.1 - Times-to-Failure Data with Suspensions}}}\,</math></center>
<center><math>\begin{matrix}
\text{Data Point Index} & \text{State F or S} & \text{State End Time} \\
\text{1} & \text{F} & \text{8} \\
\text{2} & \text{F} & \text{10} \\
\text{3} & \text{F} & \text{15} \\
\text{4} & \text{F} & \text{17} \\
\text{5} & \text{F} & \text{19} \\
\text{6} & \text{F} & \text{26} \\
\text{7} & \text{F} & \text{27} \\
\text{8} & \text{S} & \text{28} \\
\text{9} & \text{S} & \text{28} \\
\text{10} & \text{S} & \text{28} \\
\end{matrix}</math></center>

:• Determine the valve's design life if specifications call for a reliability goal of 0.90.
:• The valve is to be used in a pumping device that requires 1 month of continuous operation. What is the probability of the pump failing due to the valve?

This data set can be entered into Weibull++ as follows:

[[Image:sof-folio.png|thumb|center|400px| ]]

The computed parameters for maximum likelihood are:

::<math>\begin{align}
& \widehat{\mu }= & 22.34 \\
& \hat{\sigma }= & 6.15
\end{align}</math>

:• The valve's design life, along with 90% two sided confidence bounds, can be obtained using the QCP as follows:

[[Image:ldaLD10.6.gif|thumb|center|300px| ]]

:• The probability, along with 90% two sided confidence bounds, that the pump fails due to a valve failure during the first month is obtained as follows:

[[Image:ldaLD10.7.gif|thumb|center|300px| ]]