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We consider a $l_1$-penalization procedure in the non-parametric Gaussian regression model. In many concrete examples, the dimension $d$ of the input variable $X$ is very large (sometimes depending on the number of observations). Estimation of a $beta$-regular regression function $f$ cannot be faster than the slow rate $n^{-2beta/(2beta+d)}$. Hopefully, in some situations, $f$ depends only on a few numbers of the coordinates of $X$. In this paper, we construct two procedures. The first one selects, with high probability, these coordinates. Then, using this subset selection method, we run a local polynomial estimator (on the set of interesting coordinates) to estimate the regression function at the rate $n^{-2beta/(2beta+d^*)}$, where $d^*$, the real dimension of the problem (exact number of variables whom $f$ depends on), has replaced the dimension $d$ of the design. To achieve this result, we used a $l_1$ penalization method in this non-parametric setup.
This was a revision of arXiv:1105.2454v1 from 2012. It considers a variation on the STIV estimator where, instead of one conic constraint, there are as many conic constraints as moments (instruments) allowing to use more directly moderate deviations
In this paper, we consider regression models with a Hilbert-space-valued predictor and a scalar response, where the response depends on the predictor only through a finite number of projections. The linear subspace spanned by these projections is cal
Meinshausen and Buhlmann [Ann. Statist. 34 (2006) 1436--1462] showed that, for neighborhood selection in Gaussian graphical models, under a neighborhood stability condition, the LASSO is consistent, even when the number of variables is of greater ord
Suppose that $Y$ is a scalar and $X$ is a second-order stochastic process, where $Y$ and $X$ are conditionally independent given the random variables $xi_1,...,xi_p$ which belong to the closed span $L_X^2$ of $X$. This paper investigates a unified fr
We study the problem of high-dimensional variable selection via some two-step procedures. First we show that given some good initial estimator which is $ell_{infty}$-consistent but not necessarily variable selection consistent, we can apply the nonne