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 $Y(t) = \langles, t\rangle + \sigmaL(t)$, for all $t \epsilon \mathbb{H}$,where $L$ is some Gaussian isonormal process. This framework allows us in particular to consider the classical “Gaussian sequence model” for which $\mathbb{H} = l_2(\mathbb{N}*)$ and $L(t) = \sum_{\lambda\geq1}t_{\lambda}\varepsilon_{\lambda}$, where $(\varepsilon_{\lambda})_{\lambda\geq1}$ 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 $\mathbb{H}$ (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 $\sigma$ tends to zero. Our construction is an alternative to the one by Efroïmovich and Low for hyperrectangles and provides new results otherwise.