Template:Loglogistic 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.

Bounds on the Parameters

The lower and upper bounds [math]\displaystyle{ {\mu } }[/math] , are estimated from:

[math]\displaystyle{ \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]\displaystyle{ {{\widehat{\sigma }}} }[/math] , [math]\displaystyle{ \ln ({{\widehat{\sigma }}}) }[/math] is treated as normally distributed, and the bounds are estimated from:

[math]\displaystyle{ \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]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds.

The variances and covariances of [math]\displaystyle{ \widehat{\mu } }[/math] and [math]\displaystyle{ \widehat{\sigma } }[/math] are estimated as follows:

[math]\displaystyle{ \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]\displaystyle{ \Lambda }[/math] is the log-likelihood function of the loglogistic distribution.

Bounds on Reliability

The reliability of the logistic distribution is:

[math]\displaystyle{ \widehat{R}=\frac{1}{1+\exp (\widehat{z})} }[/math]

where:

[math]\displaystyle{ \widehat{z}=\frac{{t}'-\widehat{\mu }}{\widehat{\sigma }} }[/math]

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

[math]\displaystyle{ {{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} }[/math]

[math]\displaystyle{ {{z}_{L}}=\widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})\text{ }}\text{ } }[/math]

where:

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ {{R}_{U}}=\frac{1}{1+{{e}^{{{z}_{L}}}}}\text{(Upper bound)} }[/math]

[math]\displaystyle{ {{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]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })={{e}^{\widehat{\mu }+\widehat{\sigma }z}} }[/math]

where:

[math]\displaystyle{ z=\ln (1-R)-\ln (R) }[/math]

or:

[math]\displaystyle{ \ln (T)=\widehat{\mu }+\widehat{\sigma }z }[/math]

Let:

[math]\displaystyle{ u=\ln (T)=\widehat{\mu }+\widehat{\sigma }z }[/math]

then:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ } }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ } }[/math]

where:

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ {{T}_{U}}={{e}^{{{u}_{U}}}}\text{ (upper bound)} }[/math]

[math]\displaystyle{ {{T}_{L}}={{e}^{{{u}_{L}}}}\text{ (lower bound)} }[/math]