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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.
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| {{loglogistic probability density function}}
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| {{loglogistic mean median and mode}}
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| {{loglogistic standard deviation}}
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| {{loglogistic reliability function}}
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| {{loglogistic reliable life}}
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| {{loglogistic failure rate function}}
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| {{loglogistic distribution characteristics}}
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| {{loglogistic confidence bounds}}
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| ====Bounds on Time====
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| 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:
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| ::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })={{e}^{\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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| :or:
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| ::<math>\ln (T)=\widehat{\mu }+\widehat{\sigma }z</math>
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| :Let:
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| ::<math>u=\ln (T)=\widehat{\mu }+\widehat{\sigma }z</math>
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| :then:
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| ::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>
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| ::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>
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| :where:
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| ::<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>
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| :or:
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| ::<math>Var(\widehat{u})=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}}={{e}^{{{u}_{U}}}}\text{ (upper bound)}</math>
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|
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| ::<math>{{T}_{L}}={{e}^{{{u}_{L}}}}\text{ (lower bound)}</math>
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| ====A LogLogistic Distribution Example====
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| Determine the loglogistic parameter estimates for the data given in Table 10.3.
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| <center><math>\overset{{}}{\mathop{\text{Table 10}\text{.3 - Test data}}}\,</math></center>
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| <center><math>\begin{matrix}
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| \text{Data point index} & \text{Last Inspected} & \text{State End time} \\
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| \text{1} & \text{105} & \text{106} \\
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| \text{2} & \text{197} & \text{200} \\
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| \text{3} & \text{297} & \text{301} \\
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| \text{4} & \text{330} & \text{335} \\
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| \text{5} & \text{393} & \text{401} \\
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| \text{6} & \text{423} & \text{426} \\
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| \text{7} & \text{460} & \text{468} \\
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| \text{8} & \text{569} & \text{570} \\
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| \text{9} & \text{675} & \text{680} \\
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| \text{10} & \text{884} & \text{889} \\
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| \end{matrix}</math></center>
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| 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:
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| ::<math>\begin{align}
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| & {{{\hat{\mu }}}^{\prime }}= & 5.9772 \\
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| & {{{\hat{\sigma }}}_{{{T}'}}}= & 0.3256
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| \end{align}</math>
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| For rank regression on <math>X\ \ :</math>
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| ::<math>\begin{align}
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| & \hat{\mu }= & 5.9281 \\
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| & \hat{\sigma }= & 0.3821
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| \end{align}</math>
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| For rank regression on <math>Y\ \ :</math>
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| ::<math>\begin{align}
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| & \hat{\mu }= & 5.9772 \\
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| & \hat{\sigma }= & 0.3256
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| \end{align}</math>
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