Adaptive estimation of a quadratic functional by model selection

Abstract

We consider the problem of estimating ∥s∥2 when s belongs to some separable Hubert space and one observes the Gaussian process Y(t) = (s, t) + σL(t), for all t ∈ ℍ, where L is some Gaussian isonormal process. This framework allows us in particular to consider the classical "Gaussian sequence model" for which H = l2(ℕ*) and L(t) = Σλ≥1tλελ, where (ελ)λ≥1 is a sequence of i.i.d. standard normal variables. Our approach consists in considering some at most countable families of finite-dimensional linear subspaces of ℍ (the models) and then using model selection via some conveniently penalized least squares criterion to build new estimators of ∥s∥2. We prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s, depending on the family of models from which they are built. In particular, in the context of the Gaussian sequence model, a convenient choice of the family of models allows denning estimators which are adaptive over collections of hyperrectangles, ellipsoids, lp-bodies or Besov bodies. We take special care to describe the conditions under which the penalized estimator is efficient when the level of noise σ tends to zero. Our construction is an alternative to the one by Efroïmovich and Low for hyperrectangles and provides new results otherwise.