Reliability Decision Problems under Conditions of Ageing

Abstract

Abstract Wear-out and infant mortality of items are fundamental concepts in the statistical analysis of failure and survival data. Our aim is the study of some general qualitative properties of Bayes decision procedures when statistical data are failure and survival times of units which undergo wear-out or infant mortality. In particular we obtain ordering properties of the acceptance regions in the “predictive” two-action decision problem, by using the notion of majorization. A prerequisite for our study is an appropriate formulation of the concept of wear-out from the Bayesian viewpoint: in the frequentist approach, situations of wear-out are modeled by the Increasing Failure Rate (IFR) property of the distributions of lifetimes of units; this characterization is no longer valid when probability is intended as a degree of belief. Indeed, even if wear-out is present, it may happen that, while progressively observing the survival of a unit, we become more and more optimistic about its residual lifetime, at least in an initial period. This happens when our initial state of information (about possible unknown parameters affecting the random mechanism of failure) is too pessimistic. In a recent paper, Barlow and Mendel gave a suitable formulation of wear-out (Schur-concavity of the joint survival function). We shall use this as a general condition for our results after showing that it is equivalent to a significant property of conditional survival probabilities for residual lifetimes, given the data about already observed survivals. An analysis of the particular case of “Schur-constant” survival functions will allow us to clarify the motivation for the present paper.