Template:Bounds on beta camsaa-gd - Revision history

Richard House: Redirected page to Crow-AMSAA - NHPP#Bounds on .CE.B2 2

2012-08-24T03:44:52Z

Redirected page to Crow-AMSAA - NHPP#Bounds on .CE.B2 2

← Older revision		Revision as of 03:44, 24 August 2012
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	~~====Bounds on <math>\beta </math>====~~		#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on__.CE.B2_2]]
	~~=====Fisher Matrix Bounds=====~~
	~~The parameter <math>\beta </math> must be positive, thus <math>\ln \beta </math> is treated as being normally distributed as well.~~

	~~::<math>\frac{\ln \hat{\beta }-\ln \beta }{\sqrt{Var(\ln \hat{\beta }})}\ \tilde{\ }\ N(0,1)</math>~~

	~~The approximate confidence bounds are given as:~~

	~~::<math>C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}</math>~~

	::<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> .
	~~<br>~~
	~~All variance can be calculated using the Fisher Matrix:~~

	~~::<math>\left~~[ ~~\begin{matrix}~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } \\~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} \\~~
	~~\end{matrix} \right]_{\beta =\widehat{\beta },\lambda =\widehat{\lambda }}^{-1}=\left~~[ ~~\begin{matrix}~~
	~~Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda }) \\~~
	~~Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta }) \\~~
	~~\end{matrix} \right]</math>~~

	~~<br>~~
	~~<math>\Lambda </math> is the natural log~~-~~likelihood function where ln <math>^{2}T={{\left( \ln T \right)}^{2}}</math> and:~~

	~~::<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>~~

	~~::<math>\begin{align}~~
	~~& \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}= & -\frac{n}{{{\lambda }^{2}}} \\~~
	~~& \frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}= & \underset{i=1}{\overset{k}{\mathop \sum }}\,\left[ \begin{matrix}~~
	{{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) \\
	~~-\left( \lambda T_{i}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i}}-\lambda T_{i-1}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i-1}} \right) \\~~
	~~\end{matrix} \right] \\~~
	~~& \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }= & -T_{K}^{\beta }\ln {{T}_{k}}~~
	~~\end{align}</math>~~

	~~=====Crow Bounds=====~~
	~~:Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K</math>~~ .
	~~:Step 2: Calculate:~~

	~~::<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>~~

	: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:

Lisa Hacker: /* Bounds on \beta */

2012-08-23T22:58:15Z

Bounds on \beta

← Older revision		Revision as of 22:58, 23 August 2012
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	===Bounds on <math>\beta </math>===		====Bounds on <math>\beta </math>====
	====Fisher Matrix Bounds====		=====Fisher Matrix Bounds=====
	The parameter <math>\beta </math> must be positive, thus <math>\ln \beta </math> is treated as being normally distributed as well.		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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	\end{align}</math>		\end{align}</math>

	====Crow Bounds====		=====Crow Bounds=====
	:Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K</math> .		:Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K</math> .
	:Step 2: Calculate:		:Step 2: Calculate:

Nicolette Young: Created page with '===Bounds on $\beta$ === ====Fisher Matrix Bounds==== The parameter $\beta$ must be positive, thus $\ln \beta$ is treated as being normall…'

2012-01-05T23:06:49Z

Created page with '===Bounds on <math>\beta </math>=== ====Fisher Matrix Bounds==== The parameter <math>\beta </math> must be positive, thus <math>\ln \beta </math> is treated as being normall…'

New page

===Bounds on <math>\beta </math>===
====Fisher Matrix Bounds====
The parameter <math>\beta </math> must be positive, thus <math>\ln \beta </math> is treated as being normally distributed as well.

::<math>\frac{\ln \hat{\beta }-\ln \beta }{\sqrt{Var(\ln \hat{\beta }})}\ \tilde{\ }\ N(0,1)</math>

The approximate confidence bounds are given as:

::<math>C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}</math>

::<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> .
<br>
All variance can be calculated using the Fisher Matrix:

::<math>\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} \\
\end{matrix} \right]_{\beta =\widehat{\beta },\lambda =\widehat{\lambda }}^{-1}=\left[ \begin{matrix}
Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda }) \\
Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta }) \\
\end{matrix} \right]</math>

<br>
<math>\Lambda </math> is the natural log-likelihood function where ln <math>^{2}T={{\left( \ln T \right)}^{2}}</math> and:

::<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>

::<math>\begin{align}
& \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}= & -\frac{n}{{{\lambda }^{2}}} \\
& \frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}= & \underset{i=1}{\overset{k}{\mathop \sum }}\,\left[ \begin{matrix}
{{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) \\
-\left( \lambda T_{i}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i}}-\lambda T_{i-1}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i-1}} \right) \\
\end{matrix} \right] \\
& \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }= & -T_{K}^{\beta }\ln {{T}_{k}}
\end{align}</math>

====Crow Bounds====
:Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K</math> .
:Step 2: Calculate:

::<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>

: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:

Template:Bounds on beta camsaa-gd - Revision history

Richard House: Redirected page to Crow-AMSAA - NHPP#Bounds on .CE.B2 2

Lisa Hacker: /* Bounds on \beta */

Nicolette Young: Created page with '===Bounds on \beta === ====Fisher Matrix Bounds==== The parameter \beta must be positive, thus \ln \beta is treated as being normall…'

Nicolette Young: Created page with '===Bounds on $\beta$ === ====Fisher Matrix Bounds==== The parameter $\beta$ must be positive, thus $\ln \beta$ is treated as being normall…'