标题： 分位数回归中的多尺度渐近正态性：Hilbert矩阵与多项式设计 显示英文标题

标题： Multiscale Asymptotic Normality in Quantile Regression: Hilbert Matrices and Polynomial Designs

摘要： 本文研究了线性模型中分位数回归估计量的渐近性质，重点关注多项式回归变量以及对重尾噪声的鲁棒性。 在误差独立同分布（i.i.d.），且在感兴趣的分位数附近具有连续密度的情况下，我们利用归一化因子$\Delta_n^{-1}$建立了分位数回归估计量的一般中心极限定理（CLT），得到渐近正态性，方差为$\tau(1-\tau)/f^2(0) \cdot D_0^{-1}$。 对于多项式回归变量的特殊情况，我们证明了设计结构会在渐近协方差中引入 Hilbert 矩阵，并推导出每个系数的显式缩放率。 这将 Pollard 和 Koenker 在最小绝对偏差（LAD）回归中的早期结果推广到任意分位数水平$\tau \in (0, 1)$。 我们还分析了估计量的收敛行为，并基于理论包含原则提出了标准 CLT 置信区间的松弛形式。 这种松弛用$T^\alpha$替换了通常的$T^{j+1/2}$缩放方式，其中$\alpha < j + 1/2$，以改善有限样本覆盖率。 通过在拉普拉斯、高斯和柯西噪声下的大量模拟验证了该方法，并突出了松弛置信区间在鲁棒性和经验准确性方面的改进。 本研究为结构化回归变量和重尾扰动下的分位数回归提供了统一的理论框架和实用推断工具。 显示英文摘要

摘要： This paper investigates the asymptotic properties of quantile regression estimators in linear models, with a particular focus on polynomial regressors and robustness to heavy-tailed noise. Under independent and identically distributed (i.i.d.) errors with continuous density around the quantile of interest, we establish a general Central Limit Theorem (CLT) for the quantile regression estimator under normalization using $\Delta_n^{-1}$, yielding asymptotic normality with variance $\tau(1-\tau)/f^2(0) \cdot D_0^{-1}$. In the specific case of polynomial regressors, we show that the design structure induces a Hilbert matrix in the asymptotic covariance, and we derive explicit scaling rates for each coefficient. This generalizes Pollard's and Koenker's earlier results on LAD regression to arbitrary quantile levels $\tau \in (0, 1)$. We also examine the convergence behavior of the estimators and propose a relaxation of the standard CLT-based confidence intervals, motivated by a theoretical inclusion principle. This relaxation replaces the usual $T^{j+1/2}$ scaling with $T^\alpha$, for $\alpha < j + 1/2$, to improve finite-sample coverage. Through extensive simulations under Laplace, Gaussian, and Cauchy noise, we validate this approach and highlight the improved robustness and empirical accuracy of relaxed confidence intervals. This study provides both a unifying theoretical framework and practical inference tools for quantile regression under structured regressors and heavy-tailed disturbances.