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| ===Bounds on <math>\lambda </math>===
| | #REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on__.CE.BB]] |
| ====Fisher Matrix Bounds====
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| The parameter <math>\lambda </math> must be positive, thus <math>\ln \lambda </math> is treated as being normally distributed as well. These bounds are based on:
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| <br>
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| ::<math>\frac{\ln \hat{\lambda }-\ln \lambda }{\sqrt{Var(\ln \hat{\lambda }})}\ \tilde{\ }\ N(0,1)</math>
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| <br>
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| The approximate confidence bounds on <math>\lambda </math> are given as:
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| <br>
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| ::<math>C{{B}_{\lambda }}=\hat{\lambda }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}}</math>
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| <br>
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| :where:
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| <br>
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| ::<math>\hat{\lambda }=\frac{n}{{{T}^{*\hat{\beta }}}}</math>
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| <br>
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| The variance calculation is the same as Eqn. (variance1).
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| | |
| ====Crow Bounds====
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| '''Time Terminated Data'''
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| <br>
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| For the 2-sided <math>(1-\alpha )</math> 100-percent confidence interval, the confidence bounds on <math>\lambda </math> are:
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| ::<math>\begin{align}
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| & {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}} \\
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| & {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2{{T}^{{\hat{\beta }}}}}
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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.
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| <br>
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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, the confidence bounds on <math>\lambda </math> are:
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| ::<math>\begin{align}
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| & {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}} \\
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| & {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}}
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| \end{align}</math>
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