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, as discussed in Meeker and Escobar [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 2-parameter distribution with parameters [math]\displaystyle{ \mu \,\! }[/math] and [math]\displaystyle{ \sigma \,\! }[/math]. The pdf 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:

[math]\displaystyle{ \tilde{T} = e^{\mu+\sigma ln(\frac{1-\sigma}{1+\sigma})}\,\! }[/math]

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{ \begin{align} {t}'=\ln (t) \end{align}\,\! }[/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{ \begin{align} {{T}_{R}}={{e}^{\mu +\sigma [\ln (1-R)-\ln (R)]}} \end{align}\,\! }[/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 pdf 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 pdf 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 pdf 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 Parameter Estimation.

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{ \begin{align} z=\ln (1-R)-\ln (R) \end{align}\,\! }[/math]

or:

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

Let:

[math]\displaystyle{ {u}=\ln (\hat{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]

General Examples

Determine the loglogistic parameter estimates for the data given in the following table.

[math]\displaystyle{ \overset{{}}{\mathop{\text{Test data}}}\,\,\! }[/math] [math]\displaystyle{ \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]

Set up the folio for times-to-failure data that includes interval and left censored data, then enter the 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]