Adaptive estimation of a quadratic functional by model selection

B Laurent, P Massart - Annals of statistics, 2000 - JSTOR
B Laurent, P Massart
Annals of statistics, 2000JSTOR
We consider the problem of estimating∥ s∥ 2 when s belongs to some separable Hilbert
space and one observes the Gaussian process, 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)=∑ λ≥ 1tλελ, where (ελ) λ≥ 1 is a
sequence of iid standard normal variables. Our approach consists in considering some at
most countable families of finite-dimensional linear subspaces of H (the models) and then …
We consider the problem of estimating ∥ s ∥2 when s belongs to some separable Hilbert space and one observes the Gaussian process , 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) = ∑λ≥ 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 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 σ tends to zero. Our construction is an alternative to the one by Efroimovich and Low for hyperrectangles and provides new results otherwise.
JSTOR
以上显示的是最相近的搜索结果。 查看全部搜索结果