Sparse quantile regression

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