Statistica Sinica 25 (2015), 1313-1335
Abstract: Failure time data collected from fielded systems provide indirect information about the performance of the system’s components. Since it is often difficult to create or simulate field conditions in laboratory settings, the process of drawing inferences about component characteristics from data on system performance is of practical importance. However, there is very little literature on this problem that treats such inferences from a nonparametric perspective, and less literature still that allows the systems of interest to be of arbitrary design. The present paper focuses on nonparametric estimation of a common component reliability function using independent samples from coherent systems of varying design whose components have independent, identically distributed lifetimes. Two estimation approaches are studied. The first is conventional, and is based on treating each of the estimation problems separately; it is shown that these mixture estimators are consistent, and their asymptotic behaviours are characterized. The second estimator is quite unconventional. It is obtained by solving multiple point-wise maximum likelihood estimation problems simultaneously, and combining the separate estimators, each at fixed time points, to obtain an overall estimator of the reliability function. We show that the latter approach produces a legitimate reliability function and that, asymptotically, it is uniformly superior to all the estimators of the first type. Related estimators of the lifetime density and failure rate functions are also obtained, and their theoretical and numerical properties are described.
Key words and phrases: Coherent system, component reliability, density estimation, failure rate estimation, inverse problem, local likelihood, maximum likelihood, nonparametric estimation, reliability polynomial, survival function, system design.