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).
- Cramer–Rao bound
- Stein phenomenon
- Malliavin calculus
- Cox model
Musta, E., Pratelli, M., & Trevisan, D. (2017). Functional Cramér–Rao bounds and Stein estimators in Sobolev spaces, for Brownian motion and Cox processes. Journal of Multivariate Analysis, 154, 135-146. https://doi.org/10.1016/j.jmva.2016.10.011