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{{template:LDABOOK|15|The LogLogistic Distribution}}
{{template:LDABOOK|15|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 [[Appendix:_Life_Data_Analysis_References|[27]]]. For example, the mathematical reasoning for the construction of the probability plotting scales is very similar for these two distributions.


===The Loglogistic Distribution===
===Loglogistic Probability Density Function===
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.
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:


====Loglogistic Probability Density Function====
::<math>f(t)=\frac{{{e}^{z}}}{\sigma {t}{{(1+{{e}^{z}})}^{2}}}\,\!</math>
The loglogistic distribution is a two-parameter distribution with parameters  <math>\mu </math>  and  <math>\sigma </math> . The  <math>pdf</math>  for this distribution is given by:
 
::<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:  
Line 20: Line 18:
   & \mu = & \text{scale parameter} \\  
   & \mu = & \text{scale parameter} \\  
  & \sigma = & \text{shape parameter}   
  & \sigma = & \text{shape parameter}   
\end{align}</math>
\end{align}\,\!</math>
 
where  <math>0<t<\infty </math> ,  <math>-\infty <\mu <\infty </math>  and  <math>0<\sigma <\infty </math> .
 
====Mean, Median and Mode====
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>
where <math>0<t<\infty \,\!</math>, <math>-\infty <\mu <\infty \,\!</math> and <math>0<\sigma <\infty \,\!</math>.


===Mean, Median and Mode===
The mean of the loglogistic distribution, <math>\overline{T}\,\!</math>, is given by:


Note that for  <math>\sigma \ge 1,</math>  <math>\overline{T}</math> does not exist.
::<math>\overline{T}={{e}^{\mu }}\Gamma (1+\sigma )\Gamma (1-\sigma )\,\!</math>


The median of the loglogistic distribution, <math>\breve{T}</math> , is given by:
Note that for <math>\sigma \ge 1,\,\!</math> <math>\overline{T}\,\!</math> does not exist.


::<math>\widehat{T}={{e}^{\mu }}</math>
The median of the loglogistic distribution, <math>\breve{T}\,\!</math>, is given by:


The mode of the loglogistic distribution,  <math>\tilde{T}</math> , if  <math>\sigma <1,</math> is given by:
::<math>\widehat{T}={{e}^{\mu }}\,\!</math>


..
The mode of the loglogistic distribution, <math>\tilde{T}\,\!</math>, if <math>\sigma <1,\,\!</math> is given by:


====The Standard Deviation====
::<math>\tilde{T} = e^{\mu+\sigma ln(\frac{1-\sigma}{1+\sigma})}\,\!</math>
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>
===The Standard Deviation===
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>


Note that for <math>\sigma \ge 0.5,</math> the standard deviation does not exist.
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 <math>T</math> , starting at age 0, for the loglogistic distribution is determined by:
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>{T}'=\ln (t)</math>
{t}'=\ln (t)
\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===
The  logistic reliable life is:
The  logistic reliable life is:


::<math>\begin{align}
{{T}_{R}}={{e}^{\mu +\sigma [\ln (1-R)-\ln (R)]}}
\end{align}\,\!</math>


::<math>{{T}_{R}}={{e}^{\mu +\sigma [\ln (1-R)-\ln (R)]}}</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==
For <math>\sigma >1\,\!</math> :


:* <math>f(t)\,\!</math> decreases monotonically and is convex. Mode and mean do not exist.


====Distribution Characteristics====
For <math>\sigma =1\,\!</math> :
For <math>\sigma >1</math> :


• <math>f(T)</math> decreases monotonically and is convex. Mode and mean do not exist.
:* <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 <math>t\to 0\,\!</math>, <math>\lambda (t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.\,\!</math>


For <math>\sigma =1</math> :
For <math>0<\sigma <1\,\!</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>  
:* 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>\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 <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> 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.


• As  <math>T\to 0</math>  ,  <math>\lambda (T)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>
[[Image:WB.15 loglogistic pdf.png|center|400px| ]]


For  <math>0<\sigma <1</math> :
==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]].


• The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
===Bounds on the Parameters===
The lower and upper bounds <math>{\mu }\,\!</math>, are estimated from:


• The  <math>pdf</math> starts at zero, increases to its mode, and decreases thereafter.
::<math>\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>


• As  <math>\mu </math> increases, while  <math>\sigma </math> is kept the same, the <math>pdf</math>  gets stretched out to the right and its height decreases, while maintaining its shape.
For paramter <math>{{\widehat{\sigma }}}\,\!</math>, <math>\ln ({{\widehat{\sigma }}})\,\!</math> is treated as normally distributed, and the bounds are estimated from:


• As  <math>\mu </math>  decreases,while  <math>\sigma </math>  is kept the same, the  ..  gets pushed in towards the left and its height increases.
::<math>\begin{align}
 
   & {{\sigma }_{U}}= & {{\widehat{\sigma }}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}\text{ (upper bound)} \\  
• <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.
  & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{{{\widehat{\sigma }}}}}}}\text{ (lower bound)}  
 
\end{align}\,\!</math>
[[File:ldaLLD10.1.gif|center]]
 
====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>{\mu }'</math> , are estimated from:
 
 
<math>\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>{{\widehat{\sigma }}_{{{T}'}}}</math> , <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})</math> is treated as normally distributed, and the bounds are estimated from:


<math>\begin{align}
where <math>{{K}_{\alpha }}\,\!</math> is defined by:
  & {{\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>{{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 <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 <math>\widehat{\mu }\,\!</math> and <math>\widehat{\sigma }\,\!</math> are estimated as follows:


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.
::<math>\left( \begin{matrix}
 
The variances and covariances of  <math>\widehat{\mu }</math>  and  <math>\widehat{\sigma }</math>  are estimated as follows:  
 
<math>\left( \begin{matrix}
   \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right)  \\
   \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)  \\
   \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) & \widehat{Var}\left( \widehat{\sigma } \right)  \\
Line 138: 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 <math>\Lambda \,\!</math> is the log-likelihood function of the loglogistic distribution.


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 <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>.


Here  <math>0<t<\infty </math> ,  <math>-\infty <\mu <\infty </math>  ,  <math>0<\sigma <\infty </math> , therefore  <math>0<\ln (t)<\infty </math>    and  <math>z</math>  also is changing from  <math>-\infty </math>  till  <math>+\infty </math>  .The bounds on <math>z</math> are estimated from:  
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:  


<math>z=\ln (1-R)-\ln (R)</math>
::<math>\begin{align}
 
z=\ln (1-R)-\ln (R)
\end{align}\,\!</math>


or:  
or:  


<math>\ln (T)=\widehat{\mu }+\widehat{\sigma }z</math>
::<math>\ln (\hat{T})=\widehat{\mu }+\widehat{\sigma }z\,\!</math>
 


Let:  
Let:  


<math>u=\ln (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>
 
 
====A LogLogistic Distribution Example====
Determine the loglogistic parameter estimates for the data given in Table 10.3.
 
<math>\overset{{}}{\mathop{\text{Table 10}\text{.3 - Test data}}}\,</math>
 
 
<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>
 
 
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>{{T}_{L}}={{e}^{{{u}_{L}}}}\text{ (lower bound)}\,\!</math>


<math>\begin{align}
==General Examples==
  & \hat{\mu }= & 5.9772 \\
{{:Loglogistic Distribution Example}}
& \hat{\sigma }= & 0.3256 
\end{align}</math>

Latest revision as of 17:12, 23 December 2015

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Chapter 15: The Loglogistic Distribution


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Chapter 15  
The Loglogistic Distribution  

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Available Software:
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More Resources:
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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.
WB.15 loglogistic pdf.png

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]
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