Adaptive estimation of a quadratic functional by model selection
We consider the problem of estimating ∥s∥ 2 when s belongs to some separable Hilbert space and one observes the Gaussian process Y(t) = (s, t) + σ L(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 = l 2 (N * ) and L(t) = Σ λ≥1 t λ e λ , where (e λ ) 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 defining estimators which are adaptive over collections of hyperrectangles, ellipsoids, l p -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 Efroimovich and Low for hyperrectangles and provides new results otherwise.