Optimal Nonparametric Estimation for Semimartingales

This paper extends 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. Let { P } be a family of probability measures such that (D,F,P) is complete , (Ft,t~O) is a standard filtration, and X=(XI,FI,t~O) is a semimartingale for every P£ { P}. For a parameter O'(t) suppose Xl =V, 0+ Aft 0 where the Vo process is predictable and locally of bounded variation and the H~ proc~ss is a local martingale. Consider estimating equations for O'(t) of the form t Jau,adMu,o=O where the aoprocess is predictable. Under regularity conditions, an o optimal form for a o in the sense of Godambe (Ann. Math. Statist. 31 (1960), 1208-11) is determined. The method is applied to cases where M is linear in 0'. It is shown that Nelson-Aalen estimate for the cumulative hazard function is optimal in Godambe's sense. A new estimate is obtained for an extended gamma process model. Semimartin gale theory is used to indicate proofs of asymptotic normality of test statistics under the null hypotheses considered.