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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]].
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| ===Bounds on the Parameters===
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| The lower and upper bounds on the location parameter <math>\widehat{\mu }</math> are estimated from
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| :
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| ::<math>{{\mu }_{U}}=\widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })\text{ }}\text{ (upper bound)}</math>
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| ::<math>{{\mu }_{L}}=\widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })\text{ }}\text{ (lower bound)}</math>
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| The lower and upper bounds on the scale parameter <math>\widehat{\sigma }</math> are estimated from:
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| ::<math>{{\sigma }_{U}}=\widehat{\sigma }{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })\text{ }}}{\widehat{\sigma }}}}(\text{upper bound})</math>
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| ::<math>{{\sigma }_{L}}=\widehat{\sigma }{{e}^{\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })\text{ }}}{\widehat{\sigma }}}}\text{ (lower bound)}</math>
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| where <math>{{K}_{\alpha }}</math> is defined by:
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| ::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>
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| 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.
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| The variances and covariances of <math>\widehat{\mu }</math> and <math>\widehat{\sigma }</math> are estimated from the Fisher matrix, as follows:
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| ::<math>\left( \begin{matrix}
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| \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) \\
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| \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) & \widehat{Var}\left( \widehat{\sigma } \right) \\
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| \end{matrix} \right)=\left( \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } \\
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| {} & {} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }^{2}}} \\
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| \end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma }}^{-1}</math>
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| <math>\Lambda </math> is the log-likelihood function of the normal distribution, described in Chapter [[Parameter Estimation]] and [[Appendix: Distribution Log-Likelihood Equations]].
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| ===Bounds on Reliability===
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| The reliability of the logistic distribution is:
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| ::<math>\widehat{R}=\frac{1}{1+{{e}^{\widehat{z}}}}</math>
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| :where:
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| ::<math>\widehat{z}=\frac{T-\widehat{\mu }}{\widehat{\sigma }}</math>
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| 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:
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| ::<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>
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| :where:
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| ::<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>
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| :or:
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| ::<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>
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| The upper and lower bounds on reliability are:
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| ::<math>{{R}_{U}}=\frac{1}{1+{{e}^{{{z}_{L}}}}}\text{(upper bound)}</math>
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| ::<math>{{R}_{L}}=\frac{1}{1+{{e}^{{{z}_{U}}}}}\text{(lower bound)}</math>
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| ===Bounds on Time===
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| The bounds around time for a given logistic percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:
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| ::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>
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| :where:
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| ::<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>
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| :or:
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| ::<math>Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>
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| The upper and lower bounds are then found by:
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| ::<math>{{T}_{U}}=\widehat{T}+{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{upper bound})</math>
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| ::<math>{{T}_{L}}=\widehat{T}-{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{lower bound})</math>
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