Template:Maximum likelihood estimators camsaa-pe

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


 * $$f({{T}_{i}}|{{T}_{i-1}})=\frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta -1}}\cdot {{e}^{-\tfrac{1}\left( T_{i}^{\beta }-T_{i-1}^{\beta } \right)}}$$

The likelihood function is:


 * $$L={{\lambda }^{n}}{{\beta }^{n}}{{e}^{-\lambda {{T}^{*\beta }}}}\underset{i=1}{\overset{n}{\mathop \prod }}\,T_{i}^{\beta -1}$$

where $${{T}^{*}}$$  is the termination time and is given by:


 * $${{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\}$$

Taking the natural log on both sides:


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

And differentiating with respect to $$\lambda $$  yields:


 * $$\frac{\partial \Lambda }{\partial \lambda }=\frac{n}{\lambda }-{{T}^{*\beta }}$$

Set equal to zero and solve for $$\lambda $$ :


 * $$\widehat{\lambda }=\frac{n}$$

Now differentiate Eqn. (amsaa4) with respect to $$\beta $$ :


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

Set equal to zero and solve for $$\beta $$ :


 * $$\widehat{\beta }=\frac{n}{n\ln {{T}^{*}}-\underset{i=1}{\overset{n}{\mathop{\sum }}}\,\ln {{T}_{i}}}$$