The Loglogistic Distribution
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
The loglogistic distribution is a two-parameter distribution with parameters [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \sigma }[/math] . The [math]\displaystyle{ pdf }[/math] for this distribution is given by:
- [math]\displaystyle{ f(T)=\frac{{{e}^{z}}}{\sigma T{{(1+{{e}^{z}})}^{2}}} }[/math]
where:
- [math]\displaystyle{ z=\frac{{T}'-\mu }{\sigma } }[/math]
- [math]\displaystyle{ {T}'=\ln (T) }[/math]
and:
- [math]\displaystyle{ \begin{align} & \mu = & \text{scale parameter} \\ & \sigma = & \text{shape parameter} \end{align} }[/math]
where [math]\displaystyle{ 0\lt t\lt \infty }[/math] , [math]\displaystyle{ -\infty \lt \mu \lt \infty }[/math] and [math]\displaystyle{ 0\lt \sigma \lt \infty }[/math] .
Mean, Median and Mode
The mean of the loglogistic distribution, [math]\displaystyle{ \overline{T} }[/math] , is given by:
- [math]\displaystyle{ \overline{T}={{e}^{\mu }}\Gamma (1+\sigma )\Gamma (1-\sigma ) }[/math]
Note that for [math]\displaystyle{ \sigma \ge 1, }[/math] [math]\displaystyle{ \overline{T} }[/math] does not exist.
The median of the loglogistic distribution, [math]\displaystyle{ \breve{T} }[/math] , is given by:
- [math]\displaystyle{ \widehat{T}={{e}^{\mu }} }[/math]
The mode of the loglogistic distribution, [math]\displaystyle{ \tilde{T} }[/math] , if [math]\displaystyle{ \sigma \lt 1, }[/math] is given by:
..
The Standard Deviation
The standard deviation of the loglogistic distribution, [math]\displaystyle{ {{\sigma }_{T}} }[/math] , is given by:
- [math]\displaystyle{ {{\sigma }_{T}}={{e}^{\mu }}\sqrt{\Gamma (1+2\sigma )\Gamma (1-2\sigma )-{{(\Gamma (1+\sigma )\Gamma (1-\sigma ))}^{2}}} }[/math]
Note that for [math]\displaystyle{ \sigma \ge 0.5, }[/math] the standard deviation does not exist.
The Loglogistic Reliability Function
The reliability for a mission of time [math]\displaystyle{ T }[/math] , starting at age 0, for the loglogistic distribution is determined by:
- [math]\displaystyle{ R=\frac{1}{1+{{e}^{z}}} }[/math]
where:
- [math]\displaystyle{ z=\frac{{T}'-\mu }{\sigma } }[/math]
- [math]\displaystyle{ {T}'=\ln (t) }[/math]
The unreliability function is:
- [math]\displaystyle{ F=\frac{{{e}^{z}}}{1+{{e}^{z}}} }[/math]
The loglogistic Reliable Life
The logistic reliable life is:
- [math]\displaystyle{ {{T}_{R}}={{e}^{\mu +\sigma [\ln (1-R)-\ln (R)]}} }[/math]
The loglogistic Failure Rate Function
The loglogistic failure rate is given by:
- [math]\displaystyle{ \lambda (T)=\frac{{{e}^{z}}}{\sigma T(1+{{e}^{z}})} }[/math]
Distribution Characteristics
For [math]\displaystyle{ \sigma \gt 1 }[/math] :
• [math]\displaystyle{ f(T) }[/math] decreases monotonically and is convex. Mode and mean do not exist.
For [math]\displaystyle{ \sigma =1 }[/math] :
• [math]\displaystyle{ f(T) }[/math] decreases monotonically and is convex. Mode and mean do not exist. As [math]\displaystyle{ T\to 0 }[/math] , [math]\displaystyle{ f(T)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}. }[/math]
• As [math]\displaystyle{ T\to 0 }[/math] , [math]\displaystyle{ \lambda (T)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}. }[/math]
For [math]\displaystyle{ 0\lt \sigma \lt 1 }[/math] :
• The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
• The [math]\displaystyle{ pdf }[/math] starts at zero, increases to its mode, and decreases thereafter.
• As [math]\displaystyle{ \mu }[/math] increases, while [math]\displaystyle{ \sigma }[/math] is kept the same, the [math]\displaystyle{ pdf }[/math] gets stretched out to the right and its height decreases, while maintaining its shape.
• As [math]\displaystyle{ \mu }[/math] decreases,while [math]\displaystyle{ \sigma }[/math] is kept the same, the .. gets pushed in towards the left and its height increases.
• [math]\displaystyle{ \lambda (T) }[/math] increases till [math]\displaystyle{ T={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}} }[/math] and decreases thereafter. [math]\displaystyle{ \lambda (T) }[/math] is concave at first, then becomes convex.
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]\displaystyle{ {\mu }' }[/math] , are estimated from:
[math]\displaystyle{ \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]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}} }[/math] , [math]\displaystyle{ \ln ({{\widehat{\sigma }}_{{{T}'}}}) }[/math] is treated as normally distributed, and the bounds are estimated from:
[math]\displaystyle{ \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]\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 \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]
A LogLogistic Distribution Example
Determine the loglogistic parameter estimates for the data given in Table 10.3.
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]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 5.9772 \\ & {{{\hat{\sigma }}}_{{{T}'}}}= & 0.3256 \end{align} }[/math]
For rank regression on [math]\displaystyle{ X\ \ : }[/math]
- [math]\displaystyle{ \begin{align} & \hat{\mu }= & 5.9281 \\ & \hat{\sigma }= & 0.3821 \end{align} }[/math]
For rank regression on [math]\displaystyle{ Y\ \ : }[/math]
- [math]\displaystyle{ \begin{align} & \hat{\mu }= & 5.9772 \\ & \hat{\sigma }= & 0.3256 \end{align} }[/math]