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====Confidence Bounds====
#REDIRECT [[The_Logistic_Distribution]]
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| ]]

Latest revision as of 08:58, 3 August 2012

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