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===The Logistic Distribution===
#REDIRECT [[The_Logistic_Distribution]]
The logistic distribution has been used for growth models, and is used in a certain type of regression known as the logistic regression. It has also applications in modeling life data. The shape of the logistic distribution and the normal distribution are very similar [27]. There are some who argue that the logistic distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small location parameter, the issue of negative failure times should not present itself as a problem.
{{logistic probability density function}}
 
{{logistic mean median and mode}}
 
 
{{logistic standard deviation}}
 
{{logistic reliability function}}
 
 
{{logistic conditional reliability function}}
 
{{logistic reliable life}}
 
{{logistic failure rate function}}
 
{{characteristics of the logistic distribution}}
 
====Weibull++ Notes on Negative Time Values====
One of the disadvantages of using the logistic distribution for reliability calculations is the fact that the logistic distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).
 
{{logistic probability paper}}
 
====Confidence Bounds====
In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. 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 location parameter  <math>\widehat{\mu }</math>  are estimated from
:
 
::<math>{{\mu }_{U}}=\widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })\text{ }}\text{ (upper bound)}</math>
 
<math>{{\mu }_{L}}=\widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })\text{ }}\text{ (lower bound)}</math>
 
The lower and upper bounds on the scale parameter  <math>\widehat{\sigma }</math>  are estimated from:
 
::<math>{{\sigma }_{U}}=\widehat{\sigma }{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })\text{ }}}{\widehat{\sigma }}}}(\text{upper bound})</math>
 
 
::<math>{{\sigma }_{L}}=\widehat{\sigma }{{e}^{\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })\text{ }}}{\widehat{\sigma }}}}\text{ (lower bound)}</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>
 
 
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 from the Fisher matrix, as follows:
 
::<math>\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>
 
 
<math>\Lambda </math>  is the log-likelihood function of the normal distribution, described in Chapter 3 and Appendix C.
 
=====Bounds on Reliability=====
The reliability of the logistic distribution is:
 
::<math>\widehat{R}=\frac{1}{1+{{e}^{\widehat{z}}}}</math>
 
:where:
 
::<math>\widehat{z}=\frac{T-\widehat{\mu }}{\widehat{\sigma }}</math>
 
 
Here  <math>-\infty <T<\infty </math> ,  <math>-\infty <\mu <\infty </math>  ,  <math>0<\sigma <\infty </math> . Therefore,  <math>z</math>  also is changing from  <math>-\infty </math>  to  <math>+\infty </math> . Then the bounds on  <math>z</math>  are estimated from:
 
::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})\text{ }}</math>
 
 
::<math>{{z}_{L}}=\widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})\text{ }}\text{ }</math>
 
:where:
 
::<math>Var(\widehat{z})={{(\frac{\partial z}{\partial \mu })}^{2}}Var(\widehat{\mu })+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>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>{{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>
 
=====Bounds on Time=====
The bounds around time for a given logistic percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:
 
::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>
 
:where:
 
::<math>z=\ln (1-R)-\ln (R)</math>
 
::<math>Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>
 
:or:
 
::<math>Var(\widehat{T})=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>{{T}_{U}}=\widehat{T}+{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{upper bound})</math>
 
::<math>{{T}_{L}}=\widehat{T}-{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{lower bound})</math>
 
=====A Logistic Distribution Example=====
The lifetime of a mechanical valve is known to follow a logistic distribution. Ten units were tested for 28 months and the following months-to-failure data was collected.
 
<center><math>\overset{{}}{\mathop{\text{Table 10}\text{.1 - Times-to-Failure Data with Suspensions}}}\,</math></center>
<center><math>\begin{matrix}
  \text{Data Point Index} & \text{State F or S} & \text{State End Time}  \\
  \text{1} & \text{F} & \text{8}  \\
  \text{2} & \text{F} & \text{10}  \\
  \text{3} & \text{F} & \text{15}  \\
  \text{4} & \text{F} & \text{17}  \\
  \text{5} & \text{F} & \text{19}  \\
  \text{6} & \text{F} & \text{26}  \\
  \text{7} & \text{F} & \text{27}  \\
  \text{8} & \text{S} & \text{28}  \\
  \text{9} & \text{S} & \text{28}  \\
  \text{10} & \text{S} & \text{28}  \\
\end{matrix}</math></center>
 
:• Determine the valve's design life if specifications call for a reliability goal of 0.90.
:• The valve is to be used in a pumping device that requires 1 month of continuous operation. What is the probability of the pump failing due to the valve?
 
This data set can be entered into Weibull++ as follows:
 
[[Image:sof-folio.png|thumb|center|400px| ]]
 
The computed parameters for maximum likelihood are:
 
::<math>\begin{align}
  & \widehat{\mu }= & 22.34 \\
& \hat{\sigma }= & 6.15 
\end{align}</math>
 
:• The valve's design life, along with 90% two sided confidence bounds, can be obtained using the QCP as follows:
 
[[Image:ldaLD10.6.gif|thumb|center|300px| ]]
 
:• The probability, along with 90% two sided confidence bounds, that the pump fails due to a valve failure during the first month is obtained as follows:
 
 
[[Image:ldaLD10.7.gif|thumb|center|300px| ]]

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