{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,22]],"date-time":"2026-10-22T14:52:54Z","timestamp":1792680774463,"version":"4.5.0"},"reference-count":34,"publisher":"Institute of Mathematical Statistics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2011,2,1]]},"abstract":"<jats:p>We consider median regression and, more generally, a possibly infinite collection of quantile regressions in high-dimensional sparse models. In these models, the number of regressors p is very large, possibly larger than the sample size n, but only at most s regressors have a nonzero impact on each conditional quantile of the response variable, where s grows more slowly than n. Since ordinary quantile regression is not consistent in this case, we consider \u21131-penalized quantile regression (\u21131-QR), which penalizes the \u21131-norm of regression coefficients, as well as the post-penalized QR estimator (post-\u21131-QR), which applies ordinary QR to the model selected by \u21131-QR. First, we show that under general conditions \u21131-QR is consistent at the near-oracle rate $\\sqrt{s\/n}\\sqrt{\\log(p\\vee n)}$, uniformly in the compact set $\\mathcal{U}\\subset(0,1)$ of quantile indices. In deriving this result, we propose a partly pivotal, data-driven choice of the penalty level and show that it satisfies the requirements for achieving this rate. Second, we show that under similar conditions post-\u21131-QR is consistent at the near-oracle rate $\\sqrt{s\/n}\\sqrt{\\log(p\\vee n)}$, uniformly over $\\mathcal{U}$, even if the \u21131-QR-selected models miss some components of the true models, and the rate could be even closer to the oracle rate otherwise. Third, we characterize conditions under which \u21131-QR contains the true model as a submodel, and derive bounds on the dimension of the selected model, uniformly over $\\mathcal{U}$; we also provide conditions under which hard-thresholding selects the minimal true model, uniformly over $\\mathcal{U}$.<\/jats:p>","DOI":"10.1214\/10-aos827","type":"journal-article","created":{"date-parts":[[2010,12,3]],"date-time":"2010-12-03T10:02:08Z","timestamp":1291370528000},"page":"82-130","source":"Crossref","is-referenced-by-count":467,"title":["\u21131-penalized quantile regression in high-dimensional sparse models"],"prefix":"10.1214","volume":"39","author":[{"given":"Alexandre","family":"Belloni","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Victor","family":"Chernozhukov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"1","doi-asserted-by":"publisher","unstructured":"[1] Belloni, A. and Chernozhukov, V. 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