Optimal subsampling for composite quantile regression in big data

The composite quantile regression (CQR) is an efficient and robust alternative to the least squares for estimating regression coefficients in a linear model. We investigate optimal subsampling for CQR with massive datasets. By establishing the consistency and asymptotic normality of the CQR estimator from a general subsampling algorithm, we derive the optimal subsampling probabilities under the L- and A-optimality criteria. The L-optimality criterion minimizes the trace of the asymptotic variance–covariance matrix of the estimator for a linearly transformed regression parameters and the A-optimality criterion minimizes that of the estimator for regression parameters. The L-optimal subsampling probabilities is easy to implement as they do not depend on the densities of the responses given covariates. Based on the L-optimal subsampling probabilities, we propose algorithms for computing the resulting estimators and their asymptotic distributions and asymptotic optimality are established. To obtain standard errors for CQR estimators without estimating the densities of the responses given the covariates, we propose an iterative subsampling procedure based on the L-optimal subsampling probabilities. The proposed methods are illustrated through numerical experiments on simulated and real datasets.