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We focus on the high dimensional linear regression $Ysimmathcal{N}(Xbeta^{*},sigma^{2}I_{n})$, where $beta^{*}inmathds{R}^{p}$ is the parameter of interest. In this setting, several estimators such as the LASSO and the Dantzig Selector are known to satisfy interesting properties whenever the vector $beta^{*}$ is sparse. Interestingly both of the LASSO and the Dantzig Selector can be seen as orthogonal projections of 0 into $mathcal{DC}(s)={betainmathds{R}^{p},|X(Y-Xbeta)|_{infty}leq s}$ - using an $ell_{1}$ distance for the Dantzig Selector and $ell_{2}$ for the LASSO. For a well chosen $s>0$, this set is actually a confidence region for $beta^{*}$. In this paper, we investigate the properties of estimators defined as projections on $mathcal{DC}(s)$ using general distances. We prove that the obtained estimators satisfy oracle properties close to the one of the LASSO and Dantzig Selector. On top of that, it turns out that these estimators can be tuned to exploit a different sparsity or/and slightly different estimation objectives.
We consider a high-dimensional regression model with a possible change-point due to a covariate threshold and develop the Lasso estimator of regression coefficients as well as the threshold parameter. Our Lasso estimator not only selects covariates b
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
In high-dimensional regression, we attempt to estimate a parameter vector ${boldsymbol beta}_0in{mathbb R}^p$ from $nlesssim p$ observations ${(y_i,{boldsymbol x}_i)}_{ile n}$ where ${boldsymbol x}_iin{mathbb R}^p$ is a vector of predictors and $y_i$
We study the asymptotic properties of bridge estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase to infinity with the sample size. We are particularly interested in the use of bridge estimators
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