Maximum Likelihood Estimation of a U-Shaped Failure Rate Function

摘要

A maximum likelihood estimate (MLE) is derived for the failure rate as a function of age based on incomplete data, assuming the failure rate function is initially decreasing and subsequently increasing, with the turning point unknown. An algorithm for computation of the MLE is developed; a small worked example is included to show in detail the steps of the algorithm. A program for machine computation is presented. The MLE is shown to be a consistent estimator. No further assumption is required as to the mathematical form of the life-length distribution (such as exponential, Wiebull, normal, etc.). Thus the model may be realistically applied in a variety of reliability situations, where the so-called 'bathtub shaped' failure rate is appropriate. In the present paper it is applied to the analysis of airplane part failure data. A real life large-scale example is presented in which the failure rate of a constant speed drive unit of a jet airplane is estimated. (Author)