Dominating the maximum likelihood estimator in predicting reliability

The reliability of a series system with n independent components are estimated. The failure distributions of these components are assumed to be Weibull with parameters α i's and β i's. It is assumed that the shape parameters β i's are known. This reduces the reliability of the system to be a function of α = ( α 1, ..., α n), ξ(α), say. The MLE of ξ(α) is derived and an estimator which dominates the MLE in terms of risk, under squared error loss, is also derived. The predicted reliability using these two estimators are computed and compared.