Efficient and superefficient estimators of filtered Poisson process intensities.

Let N^K = {N^K_t, t ∈ [0, T]} be a filtered Poisson process defined on a probability space , and let θ ≔ (θ_t, t ∈ [0, T]) be a deterministic function which is the intensity of N^K under a probability P_θ. In the present paper we prove that the natural maximum likelihood estimator (MLE) N^K is an efficient estimator for θ under P_θ. Using Malliavin calculus we construct superefficient estimators of Stein type for θ which dominate, under the usual quadratic risk, the MLE N^K. 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 N^K. [ABSTRACT FROM AUTHOR]