Order-preserving functions: Applications to majorization and order statistics

Abstract

Abstract : It has been common in the theory of reliability and its practice to assume that the life of a device is exponentially distributed (i.e. it has constant hazard or failure rate) or intuitively, that the device does not wear in service. This assumption is mathematically convenient, but it should not be employed without verification of its tenability from actual data. Natural alternatives to constant hazard rate are increasing hazard rate and increasing hazard rate average, which correspond to the intuitive concept of wear-out. Recently, methods have been developed for reliability analysis based on these alternatives. Various statistical tests of the hypothesis of constant hazard rate versus the alternatives of increasing hazard rate (IHR) or increasing hazard rate average (IHRA) have been proposed. A good test must be unbiased, i.e., must yield the conclusion of constant hazard rate more often when it is true than when it is not. Such unbiased tests are those based upon functions that preserve a certain unusual vector ordering, that is, upon functions f such that f(x sub 1,...,x sub n) < f(y sub 1,...,y sub n) whenever (x sub 1,...,x sub n) the vector ordering (y sub 1,...,y sub n). Among other results, this paper presents criteria for determining whether or not a function preserves the vector ordering, i.e. whether or not the related test of constant hazard rate versus IHR or IHRA is unbiased.