A stochastic-covariate failure model with an application to case-control analysis

A stochastic process X(t) is periodically stationary (and ergodic) if, for every k⩾1 and every (t1,…,tk) in Rk, the sequence of random vectors (X(t1+n),…,X(tk+n))n=0,+1,…, is stationary (and ergodic). For such an ergodic process, let T be a positive random variable defined on the sample space of the process, representing a time of failure. The local failure-rate function is assumed to be of the form up(x),−∞<x<∞, where p(x) is a non-negative continuous function, and u>0 is a small number, tending to 0; and, for each u, T=Tu is the corresponding failure-time. It is shown that X(Tu) and uTu have, for u→0, a limiting joint distribution and are, in fact, asymptotically independent. The marginal distributions are explicitly given. Let Y be a random variable whose distribution is the limit of that of X(Tu). Under the hypothesis that p(x) is unknown or of known functional form but with unknown parameters, it is shown how p(x) can be estimated on the basis of independent copies of the random variable Y. The results are applied to the analysis of a case-control study featuring a `markerʹ process X(t) and an `event-timeʹ T. The event in the study is considered to be particularly rare, and this is reflected in the assumption u→0. The control-distribution is identified with the average marginal distribution of the (periodically stationary) marker process X(t), and the case-distribution is identified with that of Y. The particular application is a biomedical trial to determine the risk of stroke in terms of the level of an anticoagulant in the blood of the patient.