Abstract Notions of depth in regression have been introduced and studied in the literature. Regression depth (RD) of Rousseeuw and Hubert (1999), the most famous one, is a direct extension of Tukey location depth (Tukey, 1975) to regression. Like its location counterpart, the most remarkable advantage of the notion of depth in regression is to directly introduce the maximum (or deepest) regression depth estimator (aka depth induced median) for regression parameters in a multi-dimensional setting. Classical questions for the regression depth induced median include (i) is it a consistent estimator (or rather under what sufficient conditions, it is consistent)? and (ii) is there any limiting distribution? Bai and He (1999) (BH99) pioneered an attempt to answer these questions. Under some stringent conditions on (i) the design points, (ii) the conditional distributions of y given x i , and (iii) the error distributions, BH99 proved the strong consistency of the depth induced median. Under another set of conditions, BH99 showed the existence of the limiting distribution of the estimator. This article establishes the strong consistency of the depth induced median without any of the stringent conditions in BH99, and proves the existence of the limiting distribution of the estimator by sufficient conditions and an approach different from BH99.

中文翻译：

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回归深度诱导中值的大样本特性

摘要 回归中的深度概念已经在文献中进行了介绍和研究。Rousseeuw 和 Hubert (1999) 的回归深度 (RD) 是最著名的回归深度 (RD)，它是 Tukey 位置深度 (Tukey, 1975) 对回归的直接扩展。与其位置对应物一样，回归中深度概念的最显着优势是直接为多维设置中的回归参数引入最大（或最深）回归深度估计量（又名深度诱导中值）。回归深度诱导中位数的经典问题包括（i）它是否是一致的估计量（或者更确切地说，在什么充分条件下，它是一致的）？(ii) 是否有任何限制分布？Bai and He (1999) (BH99) 率先尝试回答这些问题。在（i）设计点上的一些严格条件下，(ii) 给定 xi y 的条件分布，以及 (iii) 误差分布，BH99 证明了深度诱导中值的强一致性。在另一组条件下，BH99 显示了估计量的极限分布的存在。本文建立了 BH99 中没有任何严格条件的深度诱导中值的强一致性，并通过充分条件和不同于 BH99 的方法证明了估计量的极限分布的存在。