Estimator selection with respect to Hellinger-type risks

We observe a random measure N and aim at estimating its intensity s. This statistical framework allows to deal simultaneously with the problems of estimating a density, the marginals of a multivariate distribution, the mean of a random vector with nonnegative components and the intensity of a Poisson process. Our estimation strategy is based on estimator selection. Given a family of estimators of s based on the observation of N, we propose a selection rule, based on N as well, in view of selecting among these. Little assumption is made on the collection of estimators and their dependency with respect to the observation N need not be known. The procedure offers the possibility to deal with various problems among which model selection, convex aggregation and construction of T-estimators as studied recently in Birge (Ann Inst H Poincare Probab Stat 42(3): 273-325, 2006). For illustration, we shall consider the problems of estimation, complete variable selection and selection among linear estimators in possibly non-Gaussian regression settings.