Template:Grouped data camsaa

Grouped Data
For analyzing grouped data, we follow the same logic described previously for the Duane model. If Eqn. (amsaa2a) is linearized:


 * $$\ln [E(N(T))]=\ln \lambda +\beta \ln T$$

According to Crow [9], the likelihood function for the grouped data case, (where $${{n}_{1}},$$   $${{n}_{2}},$$   $${{n}_{3}},\ldots ,$$   $${{n}_{k}}$$  failures are observed and  $$k$$  is the number of groups), is:


 * $$\underset{i=1}{\overset{k}{\mathop \prod }}\,\underset{}{\overset{}{\mathop{\Pr }}}\,({{N}_{i}}={{n}_{i}})=\underset{i=1}{\overset{k}{\mathop \prod }}\,\frac{{{(\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })}^{{{n}_{i}}}}\cdot {{e}^{-(\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })}}}{{{n}_{i}}!}$$

And the MLE of $$\lambda $$  based on this relationship is:


 * $$\widehat{\lambda }=\frac{n}{T_{k}^{\widehat{\beta }}}$$

And the estimate of $$\beta $$  is the value  $$\widehat{\beta }$$  that satisfies:


 * $$\underset{i=1}{\overset{k}{\mathop \sum }}\,{{n}_{i}}\left[ \frac{T_{i}^{\widehat{\beta }}\ln {{T}_{i}}-T_{i-1}^{\widehat{\beta }}\ln {{T}_{i-1}}}{T_{i}^{\widehat{\beta }}-T_{i-1}^{\widehat{\beta }}}-\ln {{T}_{k}} \right]=0$$

Example 4 Consider the grouped failure times data given in Table 5.2. Solve for the Crow-AMSAA parameters using MLE.

Solution To obtain the estimator of $$\beta $$, Eqn. (vv) must be solved numerically for $$\beta $$. Using RGA, the value of $$\widehat{\beta }$$  is  $$0.6315$$. Now plugging this value into Eqn. (vv1), the estimator of $$\lambda $$  is:


 * $$\begin{align}

& \widehat{\lambda }= & \frac{11}{3,{{000}^{0.6315}}} \\ & = & 0.0701 \end{align}$$

Therefore, the intensity function becomes:


 * $$\widehat{\rho }(T)=0.0701\cdot 0.6315\cdot {{T}^{-0.3685}}$$