Adaptive estimation of a quadratic functional by model selection

• 成果类型: Article
• 署名作者: Laurent, B; Massart, P
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2000
• 页码: 1302-1338
• 关键词: integral functionals; CONVERGENCE; density; compression; bounds; SPACES; rates

摘要

We consider the problem of estimating parallel tos parallel to (2) when s belongs to some separable Hilbert space and one observes the Gaussian process Y(t) = [s, t] + sigmaL(t), for all t is an element of 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) = Sigma (lambda greater than or equal to1)t(lambda)epsilon (lambda), where (epsilon (lambda))(lambda greater than or equal to1) 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 parallel tos parallel to (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, Ip-bodies or Besov bodies. We take special care to describe the conditions under which the penalized estimator is efficient when the level of noise a tends to zero. Our construction is an alternative to the one by Efroimovich and Low for hyperrectangles and provides new results otherwise.