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

TL;DR

This paper derives Cramer-Rao bounds and explores Stein estimators for drift and intensity estimation in Brownian motion and Cox processes within fractional Sobolev spaces, revealing fundamental limits and super-efficient estimators.

Contribution

It introduces Cramer-Rao bounds in fractional Sobolev spaces for Brownian motion and Cox processes, and analyzes super-efficient Stein estimators using Malliavin calculus.

Findings

No unbiased estimators with finite risk in $H^1_0$ exist.

Cramer-Rao lower bounds are established for both estimation problems.

Super-efficient Stein estimators are studied in the Gaussian case.

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 $H^1_0\subset W^{\alpha,2}\subset L^2$. In both situations, Cramer-Rao lower bounds are obtained, entailing in particular that no unbiased estimators with finite risk in $H^1_0$ exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case).