Optimal nonparametric estimation for some semimartingale stochastic differential equations

Abstract

This paper discusses the extension of a result of Godambe on parametric estimation for discrete time stochastic processes to nonparametric estimation for the continuous time case. Following Hasminskii and Ibragimov (1980), the nonparametric problem is formulated as a parametric one but with infinite dimensional parameter. A general optimality criterion for estimating functions, based on that of Godambe [ Ann. Math. Statist. 31:1208-1211 (1960)], is formulated in the case where the parameter is an element of a Banach space; and the optimality of a generalized score function is proved under further conditions. The sense in which this theory is applicable to martingale estimating functions for α in the semimartingale stochastic differential equation model dX t = α( t) dR t + dM t,a is discussed. It is shown that the Nelson-Aalen estimate for the cumulative hazard function can be regarded as optimal in Godambe's sense. Applications to diffusion models and an extended gamma process model are given also.