Non-parametric Quantile Regression and Uniform Inference with Unknown Error Distribution*

This paper studies the non-parametric estimation and uniform inference for the conditional quantile regression function (CQRF) with covariates exposed to measurement errors. We consider the case that the distribution of the measurement error is unknown and allowed to be either ordinary or super smooth. We estimate the density of the measurement error by the repeated measurements and propose the deconvolution kernel estimator for the CQRF. We derive the uniform Bahadur representation of the proposed estimator and construct the uniform confidence bands for the CQRF, uniformly in the sense for all covariates and a set of quantile indices, and establish the theoretical validity of the proposed inference. A data-driven approach for selecting the tuning parameter is also included. Monte Carlo simulations and a real data application demonstrate the usefulness of the proposed method.

本文针对存在测量误差的协变量，研究了条件分位数回归函数（conditional quantile regression function，CQRF）的非参数估计与一致推断问题。本文考虑测量误差分布未知且可服从普通光滑或超光滑分布的情形，通过重复测量数据估计测量误差的密度，并针对CQRF提出反卷积核估计量。本文推导了所提估计量的一致Bahadur表示，针对所有协变量与一组分位数指数构建了CQRF的一致置信带，并证明了所提推断方法的理论有效性。此外，本文还提出了一种数据驱动的调优参数选择方法。蒙特卡洛模拟与真实数据集应用均验证了所提方法的实用价值。