ABSTRACT
Let 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.
Acknowledgements
The authors would like to thank the referee for several helpful corrections and suggestions that led to many improvements in the paper.