Template:Maximum likelihood estimators camsaa-pe - Revision history

Lisa Hacker: Redirected page to Crow-AMSAA - NHPP

2012-08-23T12:24:53Z

← Older revision		Revision as of 12:24, 23 August 2012
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			#REDIRECT [[Crow-AMSAA - NHPP]]
	~~===Maximum Likelihood Estimators===~~
	~~The probability density function ( <math>pdf</math> ) of the <math>{{i}^{th}}</math> event given that the <math>{{(i~~-~~1)}^{th}}</math> event occurred at <math>{{T}_{i~~-~~1}}</math> is:~~

	~~<br>~~
	::<math>f({{T}_{i}}\|{{T}_{i-1}})=\frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta -1}}\cdot {{e}^{-\tfrac{1}{{{\eta }^{\beta }}}\left( T_{i}^{\beta }-T_{i-1}^{\beta } \right)}}</math>

	~~The likelihood function is:~~

	~~<br>~~
	~~::<math>L={{\lambda }^{n}}{{\beta }^{n}}{{e}^{-\lambda {{T}^{*\beta }}}}\underset{i=1}{\overset{n}{\mathop \prod }}\,T_{i}^{\beta -1}</math>~~

	~~where <math>{{T}^{*}}</math> is the termination time and is given by:~~

	~~<br>~~
	~~::<math>{{T}^{*}}=\left\{ \begin{matrix}~~
	~~{{T}_{n}}\text{ if the test is failure terminated} \\~~
	~~T>{{T}_{n}}\text{ if the test is time terminated} \\~~
	~~\end{matrix} \right\}</math>~~

	~~Taking the natural log on both sides:~~

	~~<br>~~
	~~::<math>\Lambda =n\ln \lambda +n\ln \beta -\lambda {{T}^{*\beta }}+(\beta -1)\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln {{T}_{i}}</math>~~

	~~<br>~~
	~~And differentiating with respect to <math>\lambda </math> yields:~~

	~~<br>~~
	~~::<math>\frac{\partial \Lambda }{\partial \lambda }=\frac{n}{\lambda }-{{T}^{*\beta }}</math>~~

	~~<br>~~
	~~Set equal to zero and solve for <math>\lambda </math> :~~

	~~<br>~~
	~~::<math>\widehat{\lambda }=\frac{n}{{{T}^{*\beta }}}</math>~~

	~~<br>~~
	~~Now differentiate with respect to <math>\beta </math> :~~

	~~<br>~~
	~~::<math>\frac{\partial \Lambda }{\partial \beta }=\frac{n}{\beta }-\lambda {{T}^{\beta }}\ln {{T}^{}}+\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln {{T}_{i}}</math>~~

	~~<br>~~
	~~Set equal to zero and solve for <math>\beta </math> :~~

	~~<br>~~
	~~::<math>\widehat{\beta }=\frac{n}{n\ln {{T}^{*}}-\underset{i=1}{\overset{n}{\mathop{\sum }}}\,\ln {{T}_{i}}}</math>~~

Lisa Hacker: /* Maximum Likelihood Estimators */

2012-08-23T12:23:41Z

Maximum Likelihood Estimators

← Older revision		Revision as of 12:23, 23 August 2012
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	<br>		<br>
	Now differentiate ~~Eqn. (amsaa4)~~ with respect to <math>\beta </math> :		Now differentiate with respect to <math>\beta </math> :

	<br>		<br>

Nicolette Young: Created page with ' ===Maximum Likelihood Estimators=== The probability density function ( $pdf$ ) of the ${{i}^{th}}$ event given that the ${{(i-1)}^{th}}$ eve…'

2012-01-05T21:04:18Z

Created page with ' ===Maximum Likelihood Estimators=== The probability density function ( <math>pdf</math> ) of the <math>{{i}^{th}}</math> event given that the <math>{{(i-1)}^{th}}</math> eve…'

New page

===Maximum Likelihood Estimators===
The probability density function ( <math>pdf</math> ) of the <math>{{i}^{th}}</math> event given that the <math>{{(i-1)}^{th}}</math> event occurred at <math>{{T}_{i-1}}</math> is:

 
::<math>f({{T}_{i}}|{{T}_{i-1}})=\frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta -1}}\cdot {{e}^{-\tfrac{1}{{{\eta }^{\beta }}}\left( T_{i}^{\beta }-T_{i-1}^{\beta } \right)}}</math>

The likelihood function is:

 
::<math>L={{\lambda }^{n}}{{\beta }^{n}}{{e}^{-\lambda {{T}^{*\beta }}}}\underset{i=1}{\overset{n}{\mathop \prod }}\,T_{i}^{\beta -1}</math>

where <math>{{T}^{*}}</math> is the termination time and is given by:

 
::<math>{{T}^{*}}=\left\{ \begin{matrix}
{{T}_{n}}\text{ if the test is failure terminated} \\
T>{{T}_{n}}\text{ if the test is time terminated} \\
\end{matrix} \right\}</math>

Taking the natural log on both sides:

 
::<math>\Lambda =n\ln \lambda +n\ln \beta -\lambda {{T}^{*\beta }}+(\beta -1)\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln {{T}_{i}}</math>

 
And differentiating with respect to <math>\lambda </math> yields:

 
::<math>\frac{\partial \Lambda }{\partial \lambda }=\frac{n}{\lambda }-{{T}^{*\beta }}</math>

 
Set equal to zero and solve for <math>\lambda </math> :

 
::<math>\widehat{\lambda }=\frac{n}{{{T}^{*\beta }}}</math>

 
Now differentiate Eqn. (amsaa4) with respect to <math>\beta </math> :

 
::<math>\frac{\partial \Lambda }{\partial \beta }=\frac{n}{\beta }-\lambda {{T}^{*\beta }}\ln {{T}^{*}}+\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln {{T}_{i}}</math>

 
Set equal to zero and solve for <math>\beta </math> :

 
::<math>\widehat{\beta }=\frac{n}{n\ln {{T}^{*}}-\underset{i=1}{\overset{n}{\mathop{\sum }}}\,\ln {{T}_{i}}}</math>

Template:Maximum likelihood estimators camsaa-pe - Revision history

Lisa Hacker: Redirected page to Crow-AMSAA - NHPP

Lisa Hacker: /* Maximum Likelihood Estimators */

Nicolette Young: Created page with ' ===Maximum Likelihood Estimators=== The probability density function ( pdf ) of the {{i}^{th}} event given that the {{(i-1)}^{th}} eve…'

Nicolette Young: Created page with ' ===Maximum Likelihood Estimators=== The probability density function ( $pdf$ ) of the ${{i}^{th}}$ event given that the ${{(i-1)}^{th}}$ eve…'