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| ===Bounds on <math>\beta </math>===
| | #REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on__.CE.B2_2]] |
| ====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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| ::<math>\frac{\ln \hat{\beta }-\ln \beta }{\sqrt{Var(\ln \hat{\beta }})}\ \tilde{\ }\ N(0,1)</math>
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| The approximate confidence bounds are given as:
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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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| ::<math>\widehat{\beta }</math> can be obtained by <math>\underset{i=1}{\overset{K}{\mathop{\sum }}}\,{{n}_{i}}\left( \tfrac{T_{i}^{{\hat{\beta }}}\ln {{T}_{i}}-T_{i-1}^{{\hat{\beta }}}\ln \,{{T}_{i-1}}}{T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}}}-\ln {{T}_{k}} \right)=0</math> .
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| <br>
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| All variance can be calculated using the Fisher Matrix:
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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 where ln <math>^{2}T={{\left( \ln T \right)}^{2}}</math> and:
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| ::<math>\Lambda =\underset{i=1}{\overset{k}{\mathop \sum }}\,\left[ {{n}_{i}}\ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-(\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{n}_{i}}! \right]</math>
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| ::<math>\begin{align}
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| & \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}= & -\frac{n}{{{\lambda }^{2}}} \\
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| & \frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}= & \underset{i=1}{\overset{k}{\mathop \sum }}\,\left[ \begin{matrix}
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| {{n}_{i}}\left( \tfrac{(T_{i}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i}}-T_{i-1}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i-1}})(T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}})-{{\left( T_{i}^{{\hat{\beta }}}\ln {{T}_{i}}-T_{i-1}^{{\hat{\beta }}}\ln {{T}_{i-1}} \right)}^{2}}}{{{(T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}})}^{2}}} \right) \\
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| -\left( \lambda T_{i}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i}}-\lambda T_{i-1}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i-1}} \right) \\
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| \end{matrix} \right] \\
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| & \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }= & -T_{K}^{\beta }\ln {{T}_{k}}
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| \end{align}</math>
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| ====Crow Bounds====
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| :Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K</math> .
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| :Step 2: Calculate:
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| ::<math>A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{[P{{(i)}^{{\hat{\beta }}}}\ln P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{\widehat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}}]}^{2}}}{[P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}}]}</math>
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| :Step 3: Calculate <math>c=\tfrac{1}{\sqrt{A}}</math> and <math>S=\tfrac{({{z}_{1-\alpha /2}})\cdot C}{\sqrt{N}}</math> . Thus an approximate 2-sided <math>(1-\alpha )</math> 100-percent confidence interval on <math>\widehat{\beta }</math> is:
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