## The Annals of Statistics

### Adaptive estimation in autoregression or -mixing regression via model selection

#### Abstract

We study the problem of estimatingsome unknown regression function in a $\beta$-mixing dependent framework. To this end, we consider some collection of models which are finite dimensional spaces. A penalized least-squares estimator (PLSE) is built on a data driven selected model among this collection. We state non asymptotic risk bounds for this PLSE and give several examples where the procedure can be applied (autoregression, regression with arithmetically $\beta$-mixing design points, regression with mixing errors, estimation in additive frameworks, estimation of the order of the autoregression). In addition we show that under a weak moment condition on the errors, our estimator is adaptive in the minimax sense simultaneously over some family of Besov balls.

#### Article information

Source
Ann. Statist., Volume 29, Issue 3 (2001), 839-875.

Dates
First available in Project Euclid: 24 December 2001

https://projecteuclid.org/euclid.aos/1009210692

Digital Object Identifier
doi:10.1214/aos/1009210692

Mathematical Reviews number (MathSciNet)
MR1865343

Zentralblatt MATH identifier
1012.62034

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62J02.

#### Citation

Baraud, Y; Comte, F.; Viennet, G. Adaptive estimation in autoregression or -mixing regression via model selection. Ann. Statist. 29 (2001), no. 3, 839--875. doi:10.1214/aos/1009210692. https://projecteuclid.org/euclid.aos/1009210692

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