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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}}
 
{{loglogistic confidence bounds}}
 
====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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