Template:Non-parametric LDA confidence bounds: Difference between revisions

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===Non-parametric Confidence Bounds===
#REDIRECT [[Non-Parametric Life Data Analysis]]
Confidence bounds for nonparametric reliability estimates can be calculated using a method similar to that of parametric confidence bounds. The difficulty in dealing with nonparametric data lies in the estimation of the variance. To estimate the variance for nonparametric data, Weibull++ uses Greenwood's formula [27]:
 
::<math>\widehat{Var}(\widehat{R}({{t}_{i}}))={{\left[ \widehat{R}({{t}_{i}}) \right]}^{2}}\cdot \underset{j=1}{\overset{i}{\mathop \sum }}\,\frac{\tfrac{{{r}_{j}}}{{{n}_{j}}}}{{{n}_{j}}\cdot \left( 1-\tfrac{{{r}_{j}}}{{{n}_{j}}} \right)}</math>
 
:where:
 
::<math>\begin{align}
  & m= & \text{ the total number of intervals} \\
& n= & \text{ the total number of units} 
\end{align}</math>
 
The variable  <math>{{n}_{i}}</math>  is defined by:
 
::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m</math>
 
:where:
 
::<math>\begin{align}
  & {{r}_{j}}= & \text{the number of failures in interval }j \\
& {{s}_{j}}= & \text{the number of suspensions in interval }j 
\end{align}</math>
 
Once the variance has been calculated, the standard error can be determined by taking the square root of the variance:
 
::<math>{{\widehat{se}}_{\widehat{R}}}=\sqrt{\widehat{Var}(\widehat{R}({{t}_{i}}))}</math>
 
This information can then be applied to determine the confidence bounds:
 
::<math>\left[ LC{{B}_{\widehat{R}}},\text{ }UC{{B}_{\widehat{R}}} \right]=\left[ \frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\cdot w},\text{ }\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})/w} \right]</math>
 
:where:
 
::<math>w={{e}^{{{z}_{\alpha }}\cdot \tfrac{{{\widehat{se}}_{\widehat{R}}}}{\left[ \widehat{R}\cdot (1-\widehat{R}) \right]}}}</math>
 
and  <math>\alpha </math>  is the desired confidence level for the 1-sided confidence bounds.
 
====  Example 12====
Determine the 1-sided confidence bounds for the reliability estimates in Example 11, with a 95% confidence level.
=====  Solution to Example 12=====
Once again, this type of problem is most readily solved by constructing a table similar to the following:
 
[[Image:lda22.1.gif|thumb|center|600px| ]]
 
The following plot illustrates these results graphically:
 
[[Image:lda22.2.gif|thumb|center|400px| ]]
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Latest revision as of 07:35, 29 June 2012

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