## Adaptive estimation of a quadratic functional by model selection.(English)Zbl 1105.62328

Summary: We consider the problem of estimating $$\| s\|^2$$ when s belongs to some separable Hilbert space and one observes the Gaussian process $$Y(t) = \langle s, t\rangle + \sigma L(t)$$, for all $$t \epsilon \mathbf H$$,where $$L$$ is some Gaussian isonormal process. This framework allows us in particular to consider the classical Gaussian sequence model for which $${\mathbf H} = l_2({\mathbf N}^*)$$ 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 $$\mathbf H$$ (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.

### MSC:

 62G05 Nonparametric estimation 46N30 Applications of functional analysis in probability theory and statistics 62M09 Non-Markovian processes: estimation
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