This paper investigates the asymptotic properties of least absolute deviation (LAD) regression for linear models with polynomial regressors, highlighting its robustness against heavy-tailed noise and outliers. Assuming independent and identically distributed (i.i.d.) errors, we establish the multiscale asymptotic normality of LAD estimators. A central result is the derivation of the asymptotic precision matrix, shown to be proportional to Hilbert matrices, with the proportionality coefficient depending on the asymptotic variance of the sample median of the noise distribution. We further explore the estimator's convergence properties, both in probability and almost surely, under varying model specifications. Through comprehensive simulations, we evaluate the speed of convergence of the LAD estimator and the empirical coverage probabilities of confidence intervals constructed under different scaling factors (T 1/2 and T $\alpha$ ). These experiments incorporate a range of noise distributions, including Laplace, Gaussian, and Cauchy, to demonstrate the estimator's robustness and efficiency. The findings underscore the versatility and practical relevance of LAD regression in handling non-standard data environments. By connecting the statistical properties of LAD estimators to classical mathematical structures, such as Hilbert matrices, this study offers both theoretical insights and practical tools for robust statistical modeling.

翻译：本文研究了线性模型中多项式回归量的最小绝对偏差（LAD）回归的渐近性质，突出了其对重尾噪声和异常值的稳健性。在假设误差独立同分布的条件下，我们建立了LAD估计量的多尺度渐近正态性。一个核心结果是推导了渐近精度矩阵，该矩阵被证明与希尔伯特矩阵成比例，比例系数取决于噪声分布样本中位数的渐近方差。我们进一步探讨了在不同模型设定下，估计量在概率和几乎必然意义上的收敛性质。通过全面的模拟实验，我们评估了LAD估计量的收敛速度，以及在不同尺度因子（T^{1/2}和T^{α}）下构建的置信区间的经验覆盖概率。这些实验涵盖了包括拉普拉斯分布、高斯分布和柯西分布在内的多种噪声分布，以展示估计量的稳健性和效率。研究结果强调了LAD回归在处理非标准数据环境中的多功能性和实际相关性。通过将LAD估计量的统计性质与希尔伯特矩阵等经典数学结构相联系，本研究为稳健统计建模提供了理论见解和实用工具。