Parametric Estimation of Hazard Functions with Stochastic Covariate Processes

Abstract

Let X(t), t ≥ 0, be a real or vector valued stochastic process and T arandom killing-time of the process which generally depends on the sample function. In thecontext of survival analysis, T represents the time to a prescribed event (e.g. system failure,time of disease symptom, etc.) and X(t) is a stochastic covariate process, observed up totime T. The conditional distribution of T, given X(t), t ≥ 0, is assumed to be of a knownfunctional form with an unknown vector parameter θ; however, the distributions of X(•) arenot specified. For an arbitrary fixed α > 0 the observable data from a single realization of Tand X(•) is min(T, α), X(t), 0 ≤ t ≤ min(T, α). For n ≥ 1 the maximum likelihood estimatorof θ is based on n independent copies of the observable data. It is shown that solutions ofthe likelihood equation are consistent and asymptotically normal and efficient under specifiedregularity conditions on the hazard function associated with the conditional distribution of T.The Fisher information matrix is represented in terms of the hazard function. The form of thehazard function is very general, and is not restricted to the commonly considered cases whereit depends on X(•) only through the present point X(t). Furthermore, the process X(•) is ageneral, not necessarily Markovian process.