E¢ cient Regressions via Optimally Combining Quantile Information

We develop a generally applicable framework for constructing e¢ cient estimators of regression models via quantile regressions. The proposed method is based on optimally combining information over multiple quantiles and can be applied to a broad range of parametric and nonparametric settings. When combining information over a …xed number of quantiles, we derive an explicit upper bound on the distance between the e¢ ciency of the proposed estimator and the Fisher information. As the number of quantiles increases, the asymptotic variance of the new estimator approaches the Cramér-Rao lower bound under appropriate conditions; in the case of non-regular statistical estimation, the proposed estimator leads to super-e¢ cient estimation. We illustrate the proposed method for several widely used regression models. Both asymptotic theory and Monte Carlo experiments show the superior performance over existing methods. AMS 2000: Primary 62F10, 62G08; Secondary 62G20.