Penalization <i>versus </i>Goldenshluger-Lepski strategies in warped bases regression

Chagny, Gaëlle

doi:10.1051/ps/2011165

Penalization versus Goldenshluger-Lepski strategies in warped bases regression

Chagny, Gaëlle

ESAIM: Probability and Statistics, Tome 17 (2013), pp. 328-358.

Résumé

This paper deals with the problem of estimating a regression function f, in a random design framework. We build and study two adaptive estimators based on model selection, applied with warped bases. We start with a collection of finite dimensional linear spaces, spanned by orthonormal bases. Instead of expanding directly the target function f on these bases, we rather consider the expansion of h = f ∘ G^-1, where G is the cumulative distribution function of the design, following Kerkyacharian and Picard [Bernoulli 10 (2004) 1053-1105]. The data-driven selection of the (best) space is done with two strategies: we use both a penalization version of a “warped contrast”, and a model selection device in the spirit of Goldenshluger and Lepski [Ann. Stat. 39 (2011) 1608-1632]. We propose by these methods two functions, ĥ_l (l = 1, 2), easier to compute than least-squares estimators. We establish nonasymptotic mean-squared integrated risk bounds for the resulting estimators, ${\hat{f}}_{l} = {\hat{h}}_{l} \circ G$ f̂_l = ĥ_l°G if G is known, or ${\hat{f}}_{l} = {\hat{h}}_{l} \circ \hat{G}$ f̂_l = ĥ_l°Ĝ (l = 1,2) otherwise, where Ĝ is the empirical distribution function. We study also adaptive properties, in case the regression function belongs to a Besov or Sobolev space, and compare the theoretical and practical performances of the two selection rules.

DOI : 10.1051/ps/2011165

Classification : 62G05, 62G08
Mots clés : adaptive estimator, model selection, nonparametric regression estimation, warped bases

@article{PS_2013__17__328_0,
     author = {Chagny, Ga\"elle},
     title = {Penalization \protect\emph{versus {}Goldenshluger-Lepski} strategies in warped bases regression},
     journal = {ESAIM: Probability and Statistics},
     pages = {328--358},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2011165},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2011165/}
}

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Chagny, Gaëlle. Penalization versus Goldenshluger-Lepski strategies in warped bases regression. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 328-358. doi : 10.1051/ps/2011165. http://www.numdam.org/articles/10.1051/ps/2011165/

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