Efficient and superefficient estimators of filtered Poisson process intensities

Abstract

ABSTRACTLet NK = {NKt, t ∈ [0, T]} be a filtered Poisson process defined on a probability space , and let θ ≔ (θt, t ∈ [0, T]) be a deterministic function which is the intensity of NK under a probability Pθ. In the present paper we prove that the natural maximum likelihood estimator (MLE) NK is an efficient estimator for θ under Pθ. Using Malliavin calculus we construct superefficient estimators of Stein type for θ which dominate, under the usual quadratic risk, the MLE NK. These superefficient estimators are given under the form where F is a random variable satisfying some assumptions and is the Malliavin derivative with respect to the compensated version of NK.