Template:Maximum likelihood estimators camsaa-cd

Maximum Likelihood Estimators
This section describes procedures for estimating the parameters of the Crow-AMSAA model for success/failure data. An example is presented illustrating these concepts. The estimation procedures described below provide maximum likelihood estimates (MLEs) for the model's two parameters, $$\lambda $$  and  $$\beta $$. The MLEs for $$\lambda $$  and  $$\beta $$  allow for point estimates for the probability of failure, given by:


 * $${{\hat{f}}_{i}}=\frac{\hat{\lambda }T_{i}^-\hat{\lambda }T_{i-1}^}=\frac{\hat{\lambda }\left( T_{i}^-T_{i-1}^ \right)}$$

And the probability of success (reliability) for each configuration $$i$$  is equal to:


 * $${{\hat{R}}_{i}}=1-{{\hat{f}}_{i}}$$

The likelihood function is:
 * $$\underset{i=1}{\overset{k}{\mathop \prod }}\,\left( \begin{matrix}

{{N}_{i}} \\ {{M}_{i}} \\ \end{matrix} \right){{\left( \frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }} \right)}^}{{\left( \frac{{{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta }} \right)}^{{{N}_{i}}-{{M}_{i}}}}$$

Taking the natural log on both sides yields:


 * $$\begin{align}

& \Lambda = & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ \ln \left( \begin{matrix}  {{N}_{i}}  \\   {{M}_{i}}  \\ \end{matrix} \right)+{{M}_{i}}\left[ \ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right] \\ & & +\underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ ({{N}_{i}}-{{M}_{i}})\left[ \ln ({{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right] \end{align}$$

Taking the derivative with respect to $$\lambda $$  and  $$\beta $$  respectively, exact MLEs for  $$\lambda $$  and  $$\beta $$  are values satisfying the following two equations:


 * $$\begin{align}

& \underset{i=1}{\overset{K}{\mathop \sum }}\,{{H}_{i}}\times {{S}_{i}}= & 0 \\ & \underset{i=1}{\overset{K}{\mathop \sum }}\,{{U}_{i}}\times {{S}_{i}}= & 0 \end{align}$$


 * where:


 * $$\begin{align}

& {{H}_{i}}= & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ T_{i}^{\beta }\ln {{T}_{i}}-T_{i-1}^{\beta }\ln {{T}_{i-1}} \right] \\ & {{S}_{i}}= & \frac{\left[ \lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta } \right]}-\frac{{{N}_{i}}-{{M}_{i}}}{\left[ {{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta } \right]} \\ & {{U}_{i}}= & T_{i}^{\beta }-T_{i-1}^{\beta }\, \end{align}$$

Example 8 A one-shot system underwent reliability growth development testing for a total of 68 trials. Delayed corrective actions were incorporated after the 14th, 33rd and 48th trials. From trial 49 to trial 68, the configuration was not changed. •	Configuration 1 experienced 5 failures, •	Configuration 2 experienced 3 failures, •	Configuration 3 experienced 4 failures and •	Configuration 4 experienced 4 failures.
 * 1)	Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
 * 2)	Estimate the unreliability and reliability by configuration.

Solution
 * 1)	The solution of Eqns. (solution1) and (solution2) provides for $$\lambda $$  and  $$\beta $$  corresponding to 0.5954 and 0.7801, respectively.
 * 2)	Table 5.6 displays the results of Eqns. (ffffi) and (rrrri).

Figures 4fig816 and 4fig817 show plots of the estimated unreliability and reliability by configuration. Table 5.6 - Estimated failure probability and reliability by configuration