Efficiency of estimators for partially specified filtered models

Abstract

Let X n1 ,…, X nn be counting processes and let Y n1 ,…, Y nn be vector-valued covariate processes. Assume that the intensity processes of the X ni with respect to the filtration generated by X ni and Y ni are known up to a (possibly infinite-dimensional) parameter, but that the distribution of X ni and Y ni is unspecified otherwise. We give conditions under which the partially specified likelihood in the sense of Gill-Slud-Jacod is locally asymptotically normal. We show that the partially specified likelihood determines a covariance bound in the sense of a Hájek-LeCam convolution theorem for estimating functionals of the underlying parameter. The theorem shows that the Huffer-McKeague estimator is efficient in Aalen's additive risk model, and that the Cox estimator for the regression coefficients and a Breslow-type estimator for the integrated baseline hazard are efficient in Cox's and in Prentice and Self's proportional hazards models.