He Xuming, Pan Xiaoou, Tan Kean Ming, Zhou Wen-Xin
Department of Statistics, University of Michigan, Ann Arbor, MI, 48109, USA.
Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093, USA.
J Econom. 2023 Feb;232(2):367-388. doi: 10.1016/j.jeconom.2021.07.010. Epub 2021 Aug 24.
Quantile regression is a powerful tool for learning the relationship between a response variable and a multivariate predictor while exploring heterogeneous effects. This paper focuses on statistical inference for quantile regression in the "increasing dimension" regime. We provide a comprehensive analysis of a convolution smoothed approach that achieves adequate approximation to computation and inference for quantile regression. This method, which we refer to as turns the non-differentiable check function into a twice-differentiable, convex and locally strongly convex surrogate, which admits fast and scalable gradient-based algorithms to perform optimization, and multiplier bootstrap for statistical inference. Theoretically, we establish explicit non-asymptotic bounds on estimation and Bahadur-Kiefer linearization errors, from which we show that the asymptotic normality of the conquer estimator holds under a weaker requirement on dimensionality than needed for conventional quantile regression. The validity of multiplier bootstrap is also provided. Numerical studies confirm conquer as a practical and reliable approach to large-scale inference for quantile regression. Software implementing the methodology is available in the R package conquer.
分位数回归是一种强大的工具，用于在探索异质效应时学习响应变量与多元预测变量之间的关系。本文聚焦于“维度增加”情况下分位数回归的统计推断。我们对一种卷积平滑方法进行了全面分析，该方法在计算和分位数回归推断方面实现了充分逼近。我们将这种方法称为“征服法”，它将不可微的检验函数转化为一个二阶可微、凸且局部强凸的替代函数，这使得基于梯度的快速且可扩展的算法能够进行优化，并使用乘子自助法进行统计推断。从理论上讲，我们建立了关于估计和巴哈杜尔 - 基弗线性化误差的明确非渐近界，由此表明，与传统分位数回归相比，在对维度要求更弱的情况下，“征服”估计量的渐近正态性成立。我们还证明了乘子自助法的有效性。数值研究证实，“征服法”是分位数回归大规模推断的一种实用且可靠的方法。实现该方法的软件可在R包“conquer”中获取。
