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| | | #REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on__.CE.B2]] |
| ===Bounds on <math>\beta </math>===
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| ====Fisher Matrix Bounds====
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| The parameter <math>\beta </math> must be positive, thus <math>\ln \beta </math> is treated as being normally distributed as well.
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| <br>
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| ::<math>\frac{\ln \hat{\beta }-\ln \beta }{\sqrt{Var(\ln \hat{\beta }})}\ \tilde{\ }\ N(0,1)</math>
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| <br>
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| The approximate confidence bounds are given as:
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| <br>
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| ::<math>C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}</math>
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| <br>
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| <math>\alpha </math> in <math>{{z}_{\alpha }}</math> is different ( <math>\alpha /2</math> , <math>\alpha </math> ) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher Matrix.
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| <br>
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| ::<math>\left[ \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} \\
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| \end{matrix} \right]_{\beta =\widehat{\beta },\lambda =\widehat{\lambda }}^{-1}=\left[ \begin{matrix}
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| Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda }) \\
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| Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta }) \\
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| \end{matrix} \right]</math>
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| <br>
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| ::<math>\Lambda </math> is the natural log-likelihood function:
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| <br>
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| ::<math>\Lambda =N\ln \lambda +N\ln \beta -\lambda {{T}^{\beta }}+(\beta -1)\underset{i=1}{\overset{N}{\mathop \sum }}\,\ln {{T}_{i}}</math>
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| <br>
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| ::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{N}{{{\lambda }^{2}}}</math>
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| <br>
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| :and:
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| <br>
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| ::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{N}{{{\beta }^{2}}}-\lambda {{T}^{\beta }}{{(\ln T)}^{2}}</math>
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| <br>
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| :also:
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| <br>
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| ::<math>\frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-{{T}^{\beta }}\ln T</math>
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| ====Crow Bounds====
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| '''Time Terminated Data'''
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| For the 2-sided <math>(1-\alpha )</math> 100-percent confidence interval on <math>\beta </math> , calculate:
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| ::<math>\begin{align}
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| & {{D}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2(N-1)} \\
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| & {{D}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2(N-1)}
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| \end{align}</math>
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| The fractiles can be found in the tables of the <math>{{\chi }^{2}}</math> distribution. Thus the confidence bounds on <math>\beta </math> are:
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| ::<math>\begin{align}
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| & {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\
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| & {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta }
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| \end{align}</math>
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| <br>
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| '''Failure Terminated Data'''
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| <br>
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| For the 2-sided <math>(1-\alpha )</math> 100-percent confidence interval on <math>\beta </math> , calculate:
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| ::<math>\begin{align}
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| & {{D}_{L}}= & \frac{N\cdot \chi _{\tfrac{\alpha }{2},2(N-1)}^{2}}{2(N-1)(N-2)} \\
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| & {{D}_{U}}= & \frac{N\cdot \chi _{1-\tfrac{\alpha }{2},2(N-1)}^{2}}{2(N-1)(N-2)}
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| \end{align}</math>
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| Thus the confidence bounds on <math>\beta </math> are:
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| ::<math>\begin{align}
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| & {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\
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| & {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta }
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| \end{align}</math>
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