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) =, t + (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) = ₁t__, where (_) 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, 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 tends to zero. Our construction is an alternative to the one by Efroïmovich and Low for hyperrectangles and provides new results otherwise.