The Loglogistic Distribution: Difference between revisions
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[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]
No edit summary |
No edit summary |
||
| Line 3: | Line 3: | ||
===Loglogistic Probability Density Function=== | ===Loglogistic Probability Density Function=== | ||
The loglogistic distribution is a 2-parameter distribution with parameters | The loglogistic distribution is a 2-parameter distribution with parameters <math>\mu \,\!</math> and <math>\sigma \,\!</math>. The ''pdf'' for this distribution is given by: | ||
::<math>f(t)=\frac{{{e}^{z}}}{\sigma {t}{{(1+{{e}^{z}})}^{2}}}</math> | ::<math>f(t)=\frac{{{e}^{z}}}{\sigma {t}{{(1+{{e}^{z}})}^{2}}}\,\!</math> | ||
where: | where: | ||
::<math>z=\frac{{t}'-\mu }{\sigma }</math> | ::<math>z=\frac{{t}'-\mu }{\sigma }\,\!</math> | ||
::<math>{t}'=\ln (t)</math> | ::<math>{t}'=\ln (t)\,\!</math> | ||
and: | and: | ||
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& \mu = & \text{scale parameter} \\ | & \mu = & \text{scale parameter} \\ | ||
& \sigma = & \text{shape parameter} | & \sigma = & \text{shape parameter} | ||
\end{align}</math> | \end{align}\,\!</math> | ||
where | where <math>0<t<\infty \,\!</math>, <math>-\infty <\mu <\infty \,\!</math> and <math>0<\sigma <\infty \,\!</math>. | ||
===Mean, Median and Mode=== | ===Mean, Median and Mode=== | ||
The mean of the loglogistic distribution, | The mean of the loglogistic distribution, <math>\overline{T}\,\!</math>, is given by: | ||
::<math>\overline{T}={{e}^{\mu }}\Gamma (1+\sigma )\Gamma (1-\sigma )</math> | ::<math>\overline{T}={{e}^{\mu }}\Gamma (1+\sigma )\Gamma (1-\sigma )\,\!</math> | ||
Note that for | Note that for <math>\sigma \ge 1,\,\!</math> <math>\overline{T}\,\!</math> does not exist. | ||
The median of the loglogistic distribution, | The median of the loglogistic distribution, <math>\breve{T}\,\!</math>, is given by: | ||
::<math>\widehat{T}={{e}^{\mu }}</math> | ::<math>\widehat{T}={{e}^{\mu }}\,\!</math> | ||
The mode of the loglogistic distribution, | The mode of the loglogistic distribution, <math>\tilde{T}\,\!</math>, if <math>\sigma <1,\,\!</math> is given by: | ||
::<math>\tilde{T} = e^{\mu+\sigma ln(\frac{1-\sigma}{1+\sigma})}</math> | ::<math>\tilde{T} = e^{\mu+\sigma ln(\frac{1-\sigma}{1+\sigma})}\,\!</math> | ||
===The Standard Deviation=== | ===The Standard Deviation=== | ||
The standard deviation of the loglogistic distribution, | The standard deviation of the loglogistic distribution, <math>{{\sigma }_{T}}\,\!</math>, is given by: | ||
::<math>{{\sigma }_{T}}={{e}^{\mu }}\sqrt{\Gamma (1+2\sigma )\Gamma (1-2\sigma )-{{(\Gamma (1+\sigma )\Gamma (1-\sigma ))}^{2}}}</math> | ::<math>{{\sigma }_{T}}={{e}^{\mu }}\sqrt{\Gamma (1+2\sigma )\Gamma (1-2\sigma )-{{(\Gamma (1+\sigma )\Gamma (1-\sigma ))}^{2}}}\,\!</math> | ||
Note that for | Note that for <math>\sigma \ge 0.5,\,\!</math> the standard deviation does not exist. | ||
===The Loglogistic Reliability Function=== | ===The Loglogistic Reliability Function=== | ||
The reliability for a mission of time | The reliability for a mission of time <math>T\,\!</math>, starting at age 0, for the loglogistic distribution is determined by: | ||
::<math>R=\frac{1}{1+{{e}^{z}}}</math> | ::<math>R=\frac{1}{1+{{e}^{z}}}\,\!</math> | ||
where: | where: | ||
::<math>z=\frac{{t}'-\mu }{\sigma }</math> | ::<math>z=\frac{{t}'-\mu }{\sigma }\,\!</math> | ||
::<math>\begin{align} | ::<math>\begin{align} | ||
{t}'=\ln (t) | {t}'=\ln (t) | ||
\end{align}</math> | \end{align}\,\!</math> | ||
The unreliability function is: | The unreliability function is: | ||
::<math>F=\frac{{{e}^{z}}}{1+{{e}^{z}}}</math> | ::<math>F=\frac{{{e}^{z}}}{1+{{e}^{z}}}\,\!</math> | ||
===The loglogistic Reliable Life=== | ===The loglogistic Reliable Life=== | ||
| Line 66: | Line 66: | ||
::<math>\begin{align} | ::<math>\begin{align} | ||
{{T}_{R}}={{e}^{\mu +\sigma [\ln (1-R)-\ln (R)]}} | {{T}_{R}}={{e}^{\mu +\sigma [\ln (1-R)-\ln (R)]}} | ||
\end{align}</math> | \end{align}\,\!</math> | ||
===The loglogistic Failure Rate Function=== | ===The loglogistic Failure Rate Function=== | ||
The loglogistic failure rate is given by: | The loglogistic failure rate is given by: | ||
::<math>\lambda (t)=\frac{{{e}^{z}}}{\sigma t(1+{{e}^{z}})}</math> | ::<math>\lambda (t)=\frac{{{e}^{z}}}{\sigma t(1+{{e}^{z}})}\,\!</math> | ||
==Distribution Characteristics== | ==Distribution Characteristics== | ||
For | For <math>\sigma >1\,\!</math> : | ||
:* <math>f(t)\,\!</math> | :* <math>f(t)\,\!</math> decreases monotonically and is convex. Mode and mean do not exist. | ||
For | For <math>\sigma =1\,\!</math> : | ||
:* <math>f(t)\,\!</math> | :* <math>f(t)\,\!</math> decreases monotonically and is convex. Mode and mean do not exist. As <math>t\to 0\,\!</math>, <math>f(t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.\,\!</math> | ||
:* As | :* As <math>t\to 0\,\!</math>, <math>\lambda (t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.\,\!</math> | ||
For | For <math>0<\sigma <1\,\!</math> : | ||
:* The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution. | :* 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. | :* The ''pdf'' starts at zero, increases to its mode, and decreases thereafter. | ||
:* As | :* As <math>\mu \,\!</math> increases, while <math>\sigma \,\!</math> is kept the same, the ''pdf'' gets stretched out to the right and its height decreases, while maintaining its shape. | ||
:* As | :* As <math>\mu \,\!</math> decreases,while <math>\sigma \,\!</math> is kept the same, the ''pdf'' gets pushed in towards the left and its height increases. | ||
:* <math>\lambda (t)\,\!</math> | :* <math>\lambda (t)\,\!</math> increases till <math>t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}\,\!</math> and decreases thereafter. <math>\lambda (t)\,\!</math> is concave at first, then becomes convex. | ||
[[Image:WB.15 loglogistic pdf.png|center|350px| ]] | [[Image:WB.15 loglogistic pdf.png|center|350px| ]] | ||
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===Bounds on the Parameters=== | ===Bounds on the Parameters=== | ||
The lower and upper bounds <math>{\mu }</math> , are estimated from: | The lower and upper bounds <math>{\mu }\,\!</math>, are estimated from: | ||
::<math>\begin{align} | ::<math>\begin{align} | ||
& \mu _{U}= & {{\widehat{\mu }}}+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\ | & \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)} | & \mu _{L}= & {{\widehat{\mu }}}-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)} | ||
\end{align}</math> | \end{align}\,\!</math> | ||
For paramter | For paramter <math>{{\widehat{\sigma }}}\,\!</math>, <math>\ln ({{\widehat{\sigma }}})\,\!</math> is treated as normally distributed, and the bounds are estimated from: | ||
::<math>\begin{align} | ::<math>\begin{align} | ||
& {{\sigma }_{U}}= & {{\widehat{\sigma }}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}\text{ (upper bound)} \\ | & {{\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)} | & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{{{\widehat{\sigma }}}}}}}\text{ (lower bound)} | ||
\end{align}</math> | \end{align}\,\!</math> | ||
where | 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> | ::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})\,\!</math> | ||
If | 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 | The variances and covariances of <math>\widehat{\mu }\,\!</math> and <math>\widehat{\sigma }\,\!</math> are estimated as follows: | ||
::<math>\left( \begin{matrix} | ::<math>\left( \begin{matrix} | ||
| Line 126: | Line 126: | ||
{} & {} \\ | {} & {} \\ | ||
-\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }^{2}}} \\ | -\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> | \end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma }}^{-1}\,\!</math> | ||
where | where <math>\Lambda \,\!</math> is the log-likelihood function of the loglogistic distribution. | ||
===Bounds on Reliability=== | ===Bounds on Reliability=== | ||
The reliability of the logistic distribution is: | The reliability of the logistic distribution is: | ||
::<math>\widehat{R}=\frac{1}{1+\exp (\widehat{z})}</math> | ::<math>\widehat{R}=\frac{1}{1+\exp (\widehat{z})}\,\!</math> | ||
where: | where: | ||
::<math>\widehat{z}=\frac{{t}'-\widehat{\mu }}{\widehat{\sigma }}</math> | ::<math>\widehat{z}=\frac{{t}'-\widehat{\mu }}{\widehat{\sigma }}\,\!</math> | ||
Here | Here <math>0<t<\infty \,\!</math>, <math>-\infty <\mu <\infty \,\!</math>, <math>0<\sigma <\infty \,\!</math>, therefore <math>0<t'=\ln (t)<\infty \,\!</math> and <math>z\,\!</math> also is changing from <math>-\infty \,\!</math> till <math>+\infty \,\!</math>. | ||
The bounds on | The bounds on <math>z\,\!</math> are estimated from: | ||
::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})}</math> | ::<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> | ::<math>{{z}_{L}}=\widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})\text{ }}\text{ }\,\!</math> | ||
where: | 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> | ::<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: | 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> | ::<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: | The upper and lower bounds on reliability are: | ||
::<math>{{R}_{U}}=\frac{1}{1+{{e}^{{{z}_{L}}}}}\text{(Upper bound)}</math> | ::<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> | ::<math>{{R}_{L}}=\frac{1}{1+{{e}^{{{z}_{U}}}}}\text{(Lower bound)}\,\!</math> | ||
===Bounds on Time=== | ===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: | 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> | ::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })={{e}^{\widehat{\mu }+\widehat{\sigma }z}}\,\!</math> | ||
where: | where: | ||
| Line 170: | Line 170: | ||
::<math>\begin{align} | ::<math>\begin{align} | ||
z=\ln (1-R)-\ln (R) | z=\ln (1-R)-\ln (R) | ||
\end{align}</math> | \end{align}\,\!</math> | ||
or: | or: | ||
::<math>\ln (\hat{T})=\widehat{\mu }+\widehat{\sigma }z</math> | ::<math>\ln (\hat{T})=\widehat{\mu }+\widehat{\sigma }z\,\!</math> | ||
Let: | Let: | ||
::<math>{u}=\ln (\hat{T})=\widehat{\mu }+\widehat{\sigma }z</math> | ::<math>{u}=\ln (\hat{T})=\widehat{\mu }+\widehat{\sigma }z\,\!</math> | ||
then: | then: | ||
::<math>{u}_{U}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math> | ::<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> | ::<math>{u}_{L}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }\,\!</math> | ||
where: | 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> | ::<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: | or: | ||
::<math>Var(\widehat{u})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math> | ::<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: | The upper and lower bounds are then found by: | ||
::<math>{{T}_{U}}={{e}^{{{u}_{U}}}}\text{ (upper bound)}</math> | ::<math>{{T}_{U}}={{e}^{{{u}_{U}}}}\text{ (upper bound)}\,\!</math> | ||
::<math>{{T}_{L}}={{e}^{{{u}_{L}}}}\text{ (lower bound)}</math> | ::<math>{{T}_{L}}={{e}^{{{u}_{L}}}}\text{ (lower bound)}\,\!</math> | ||
==General Examples== | ==General Examples== | ||
Revision as of 17:55, 26 September 2012
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.
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]