The Annals of Statistics
- Ann. Statist.
- Volume 28, Number 5 (2000), 1302-1338.
Adaptive estimation of a quadratic functional by model selection
B. Laurent and P. Massart
Abstract
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.
Article information
Source
Ann. Statist., Volume 28, Number 5 (2000), 1302-1338.
Dates
First available in Project Euclid: 12 March 2002
Permanent link to this document
https://projecteuclid.org/euclid.aos/1015957395
Digital Object Identifier
doi:10.1214/aos/1015957395
Mathematical Reviews number (MathSciNet)
MR1805785
Zentralblatt MATH identifier
1105.62328
Subjects
Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties 62J02: General nonlinear regression
Keywords
Adaptive estimation quadratic functionals model selection Besov bodies $l_p$-bodies Gaussian sequence model efficient estimation
Citation
Laurent, B.; Massart, P. Adaptive estimation of a quadratic functional by model selection. Ann. Statist. 28 (2000), no. 5, 1302--1338. doi:10.1214/aos/1015957395. https://projecteuclid.org/euclid.aos/1015957395
