Template:Maximum likelihood estimators camsaa-pe: Difference between revisions

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(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…')
 
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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 Eqn. (amsaa4) 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>

Latest revision as of 12:24, 23 August 2012

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