Functional Cramér–Rao bounds and Stein estimators in Sobolev spaces, for Brownian motion and Cox processes
Eni Musta (TU Delft - Statistics)
M. Pratelli (Università degli Studi di Pisa)
D. Trevisan (Università degli Studi di Pisa)
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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 .
In both situations, Cramér–Rao lower bounds are obtained, entailing in
particular that no unbiased estimators (not necessarily adapted) with
finite risk in exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case).