Better nonparametric confidence intervals via robust bias correction for quantile regression

In this article, we revisit the problem of how to construct better nonparametric confidence intervals for the conditional quantile function from an optimization perspective. We apply the fully data-driven bias correction procedure based on local polynomial smoothing to estimate the conditional quantile. To account for the effect of the estimated bias, we apply an asymptotic framework that the ratio of the bandwidth to the pilot bandwidth tends to some positive constant rather than zero as the sample size grows. We derive an alternative asymptotic normality of the proposed bias-corrected quantile estimator as well as a new asymptotic variance formula. Based on theoretical results, two new pointwise confidence intervals are constructed through resampling strategies. Extensive simulation studies show that our proposed confidence intervals enjoy better performance than other competitors in terms of coverage probabilities and interval lengths and are not sensitive to the choice of bandwidth. Finally, our proposed procedure is further illustrated through United States' natality birth data in 2017.