Logistic Confidence Bounds - Revision history

Lisa Hacker: Created page with "{{template:LDABOOK|14.1|Logistic 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 Confidence Bounds. ===Bounds on the Parameters=== The lower and upper bounds on the location parameter $\widehat{\mu }\,\!$ are estimated from : ::
2023-03-10T00:59:07Z

Created page with "{{template:LDABOOK|14.1|Logistic 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 Confidence Bounds. ===Bounds on the Parameters=== The lower and upper bounds on the location parameter <math>\widehat{\mu }\,\!</math> are estimated from : ::<math..."

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{{template:LDABOOK|14.1|Logistic 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 [[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 [[Parameter Estimation]] and [[Appendix:_Log-Likelihood_Equations|Appendix D]].

===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>\begin{align}
z=\ln (1-R)-\ln (R)
\end{align}\,\!</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>