Functional Cramer-Rao bounds and Stein estimators in Sobolev spaces, for Brownian motion and Cox processes

摘要

We investigate the problems of drift estimation for a shifted Brownian motionand intensity estimation for a Cox process on a finite interval $[0,T]$, whenthe risk is given by the energy functional associated to some fractionalSobolev space $H^1_0\subset W^{\alpha,2}\subset L^2$. In both situations,Cramer-Rao lower bounds are obtained, entailing in particular that no unbiasedestimators with finite risk in $H^1_0$ exist. By Malliavin calculus techniques,we also study super-efficient Stein type estimators (in the Gaussian case).