We develop uniformly valid confidence regions for a regression coefficient in a high-dimensional sparse LAD (least absolute deviation or median) regression model. The setting is one where the number of regressors p could be large in comparison to the sample size n, but only s n of them are...

We develop uniformly valid confidence regions for regression coefficients in a high-dimensional sparse least absolute deviation/median regression model. The setting is one where the number of regressors p could be large in comparison to the sample size n, but only s << n of them are needed to accurately describe the regression function. Our new methods are based on the instrumental median regression estimator that assembles the optimal estimating equation from the output of the post l1-penalized median regression and post l1-penalized least squares in an auxiliary equation. The estimating equation is immunized against non-regular estimation of nuisance part of the median regression function, in the sense of Neyman. We establish that in a homoscedastic regression model, the instrumental median regression estimator of a single regression coefficient is asymptotically root-n normal uniformly with respect to the underlying sparse model. The resulting confidence regions are valid uniformly with respect to the underlying model. We illustrate the value of uniformity with Monte-Carlo experiments which demonstrate that standard/naive post-selection inference breaks down over large parts of the parameter space, and the proposed method does not. We then generalize our method to the case where p1 > n regression coefficients...</<>

This work proposes new inference methods for the estimation of a regression coefficient of interest in quantile regression models. We consider high-dimensional models where the number of regressors potentially exceeds the sample size but a subset of them suffice to construct a reasonable...

We develop uniformly valid confidence regions for regression coefficients in a highdimensional sparse median regression model with homoscedastic errors. Our methods are based on a moment equation that is immunized against non-regular estimation of the nuisance part of the median regression...

This work proposes new inference methods for the estimation of a regression coefficient of interest in quantile regression models. We consider high-dimensional models where the number of regressors potentially exceeds the sample size but a subset of them suffice to construct a reasonable...

We consider median regression and, more generally, quantile regression in high-dimensional sparse models. In these models the overall number of regressors p is very large, possibly larger than the sample size n, but only s of these regressors have non-zero impact on the conditional quantile of...

In this paper we study post-penalized estimators which apply ordinary, unpenalized linear regression to the model selected by first-step penalized estimators, typically LASSO. It is well known that LASSO can estimate the regression function at nearly the oracle rate, and is thus hard to improve...

Quantile regression (QR) is a principal regression method for analyzing the impact of covariates on outcomes. The impact is described by the conditional quantile function and its functionals. In this paper we develop the nonparametric QR series framework, covering many regressors as a special...