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Sparse Quantile Regression

Authors: Le-Yu Chen
Date: 24 June 2020
Type: cemmap Working Paper, CWP30/20
DOI: 10.1920/wp.cem.2020.3020

Abstract

We consider both l0-penalized and l0-constrained quantile regression estimators. For the l0-penalized estimator, we derive an exponential inequality on the tail probability of excess quantile prediction risk and apply it to obtain non-asymptotic upper bounds on the mean-square parameter and regression function estimation errors. We also derive analogous results for the l0-constrained estimator. The resulting rates of convergence are minimax-optimal and the same as those for l1-penalized estimators. Further, we characterize expected Hamming loss for the l0-penalized estimator. We implement the proposed procedure via mixed integer linear programming and also a more scalable first-order approximation algorithm. We illustrate the finite-sample performance of our approach in Monte Carlo experiments and its usefulness in a real data application concerning conformal prediction of infant birth weights (with n ≈ 103 and up to p > 103). In sum, our l0-based method produces a much sparser estimator than the l1-penalized approach without compromising precision.

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