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===The Loglogistic Distribution===
#REDIRECT [[The_Loglogistic_Distribution]]
As may be indicated by the name, the loglogistic distribution has certain similarities to the logistic distribution. A random variable is loglogistically distributed if the logarithm of the random variable is logistically distributed. Because of this, there are many mathematical similarities between the two distributions [27]. For example, the mathematical reasoning for the construction of the probability plotting scales is very similar for these two distributions.
 
{{loglogistic probability density function}}
 
{{loglogistic mean median and mode}}
 
{{loglogistic standard deviation}}
 
{{loglogistic reliability function}}
 
{{loglogistic reliable life}}
 
{{loglogistic failure rate function}}
 
{{loglogistic distribution characteristics}}
 
====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 on the mean,  <math>{\mu }'</math> , are estimated from:
 
 
::<math>\begin{align}
  & \mu _{U}^{\prime }= & {{\widehat{\mu }}^{\prime }}+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\
& \mu _{L}^{\prime }= & {{\widehat{\mu }}^{\prime }}-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)} 
\end{align}</math>
 
 
For the standard deviation,  <math>{{\widehat{\sigma }}_{{{T}'}}}</math> ,  <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})</math>  is treated as normally distributed, and the bounds are estimated from:
 
::<math>\begin{align}
  & {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}\text{ (upper bound)} \\
& {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\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<\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 (T)=\widehat{\mu }+\widehat{\sigma }z</math>
 
 
:Let:
 
::<math>u=\ln (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>
 
====A LogLogistic Distribution Example====
Determine the loglogistic parameter estimates for the data given in Table 10.3.
 
<center><math>\overset{{}}{\mathop{\text{Table 10}\text{.3 - Test data}}}\,</math></center>
 
<center><math>\begin{matrix}
  \text{Data point index} & \text{Last Inspected} & \text{State End time}  \\
  \text{1} & \text{105} & \text{106}  \\
  \text{2} & \text{197} & \text{200}  \\
  \text{3} & \text{297} & \text{301}  \\
  \text{4} & \text{330} & \text{335}  \\
  \text{5} & \text{393} & \text{401}  \\
  \text{6} & \text{423} & \text{426}  \\
  \text{7} & \text{460} & \text{468}  \\
  \text{8} & \text{569} & \text{570}  \\
  \text{9} & \text{675} & \text{680}  \\
  \text{10} & \text{884} & \text{889}  \\
\end{matrix}</math></center>
 
 
Using Times-to-failure data under the Folio Data Type and the My data set contains interval and/or left censored data under Times-to-failure data options to enter the above data, the computed parameters for maximum likelihood are calculated to be:
 
::<math>\begin{align}
  & {{{\hat{\mu }}}^{\prime }}= & 5.9772 \\
& {{{\hat{\sigma }}}_{{{T}'}}}= & 0.3256 
\end{align}</math>
 
 
For rank regression on  <math>X\ \ :</math> 
 
::<math>\begin{align}
  & \hat{\mu }= & 5.9281 \\
& \hat{\sigma }= & 0.3821 
\end{align}</math>
 
 
For rank regression on  <math>Y\ \ :</math> 
 
::<math>\begin{align}
  & \hat{\mu }= & 5.9772 \\
& \hat{\sigma }= & 0.3256 
\end{align}</math>

Latest revision as of 09:58, 9 August 2012

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