Bootstrap inference for quantile-based modal regression

In this paper, we develop uniform inference methods for the conditional mode based on quantile regression. Specifically, we propose to estimate the conditional mode by minimizing the derivative of the estimated conditional quantile function defined by smoothing the linear quantile regression estimator, and develop two bootstrap methods, a novel pivotal bootstrap and the nonparametric bootstrap, for our conditional mode estimator. Building on high-dimensional Gaussian approximation techniques, we establish the validity of simultaneous confidence rectangles constructed from the two bootstrap methods for the conditional mode. We also extend the preceding analysis to the case where the dimension of the covariate vector is increasing with the sample size. Finally, we conduct simulation experiments and a real data analysis using U.S. wage data to demonstrate the finite sample performance of our inference method. The supplemental materials include the wage dataset, R codes and an appendix containing proofs of the main results, additional simulation results, discussion of model misspecification and quantile crossing, and additional details of the numerical implementation.

本文针对基于分位数回归（quantile regression）的条件众数（conditional mode）构建了统一的推断方法。具体而言，我们提出通过最小化经平滑处理的线性分位数回归估计量（linear quantile regression estimator）所定义的估计条件分位数函数（conditional quantile function）的导数，来估计条件众数，并针对该条件众数估计量提出两种自助法（bootstrap）：一种是新颖的枢轴自助法（pivotal bootstrap），另一种是非参数自助法（nonparametric bootstrap）。基于高维高斯近似（high-dimensional Gaussian approximation）技术，我们证明了由这两种自助法构建的条件众数同时置信矩形（simultaneous confidence rectangles）的有效性。我们还将前述分析拓展至协变量向量维度随样本量递增的情形。最后，我们通过模拟实验与美国工资数据（U.S. wage data）的实证分析，验证了所提推断方法的有限样本表现（finite sample performance）。补充材料包含该工资数据集、R代码（R codes），以及涵盖主要结果证明、额外模拟结果、模型设定偏误（model misspecification）与分位数交叉（quantile crossing）讨论，还有数值实现补充细节的附录。