Functional Cramér–Rao bounds and Stein estimators in Sobolev spaces, for Brownian motion and Cox processes

Abstract

We investigate the problems of drift estimation for a shifted Brownian motion and intensity estimation for a Cox process on a finite interval [ 0 , T ] , when the risk is given by the energy functional associated to some fractional Sobolev space H 0 1 ⊂ W α , 2 ⊂ L 2 . In both situations, Cramér–Rao lower bounds are obtained, entailing in particular that no unbiased estimators (not necessarily adapted) with finite risk in H 0 1 exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case).