## Electronic Journal of Statistics

### Adaptive variable selection in nonparametric sparse additive models

#### Abstract

We consider the problem of recovery of an unknown multivariate signal $f$ observed in a $d$-dimensional Gaussian white noise model of intensity $\varepsilon$. We assume that $f$ belongs to a class of smooth functions in $L_{2}([0,1]^{d})$ and has an additive sparse structure determined by the parameter $s$, the number of non-zero univariate components contributing to $f$. We are interested in the case when $d=d_{\varepsilon }\to \infty$ as $\varepsilon \to 0$ and the parameter $s$ stays “small” relative to $d$. With these assumptions, the recovery problem in hand becomes that of determining which sparse additive components are non-zero.

Attempting to reconstruct most, but not all, non-zero components of $f$, we arrive at the problem of almost full variable selection in high-dimensional regression. For two different choices of a class of smooth functions, we establish conditions under which almost full variable selection is possible, and provide a procedure that achieves this goal. Our procedure is the best possible (in the asymptotically minimax sense) for selecting most non-zero components of $f$. Moreover, it is adaptive in the parameter $s$. In addition to that, we complement the findings of [17] by obtaining an adaptive exact selector for the class of infinitely-smooth functions. Our theoretical results are illustrated with numerical experiments.

#### Article information

Source
Electron. J. Statist., Volume 11, Number 1 (2017), 2321-2357.

Dates
First available in Project Euclid: 27 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1495850627

Digital Object Identifier
doi:10.1214/17-EJS1275

Mathematical Reviews number (MathSciNet)
MR3656494

Zentralblatt MATH identifier
1365.62133

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

#### Citation

Butucea, Cristina; Stepanova, Natalia. Adaptive variable selection in nonparametric sparse additive models. Electron. J. Statist. 11 (2017), no. 1, 2321--2357. doi:10.1214/17-EJS1275. https://projecteuclid.org/euclid.ejs/1495850627

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