The maximum depth estimator (aka depth median) ($\bs{\beta}^*_{RD}$) induced from regression depth (RD) of Rousseeuw and Hubert (1999) (RH99) is one of the most prevailing estimators in regression. It possesses outstanding robustness similar to the univariate location counterpart. Indeed, $\bs{\beta}^*_{RD}$ can, asymptotically, resist up to $33\%$ contamination without breakdown, in contrast to the $0\%$ for the traditional (least squares and least absolute deviations) estimators (see Van Aelst and Rousseeuw, 2000) (VAR00)). The results from VAR00 are pioneering, yet they are limited to regression-symmetric populations (with a strictly positive density) and the $\epsilon$-contamination and maximum-bias model. With a fixed finite-sample size practice, the most prevailing measure of robustness for estimators is the finite-sample breakdown point (FSBP) (Donoho and Huber (1983)). Despite many attempts made in the literature, only sporadic partial results on FSBP for $\bs{\beta}^*_{RD}$ were obtained whereas an exact FSBP for $\bs{\beta}^*_{RD}$ remained open in the last twenty-plus years. Furthermore, is the asymptotic breakdown value $1/3$ (the limit of an increasing sequence of finite-sample breakdown values) relevant in the finite-sample practice? (Or what is the difference between the finite-sample and the limit breakdown values?). Such discussions are yet to be given in the literature. This article addresses the above issues, revealing an intrinsic connection between the regression depth of $\bs{\beta}^*_{RD}$ and the newly obtained exact FSBP. It justifies the employment of $\bs{\beta}^*_{RD}$ as a robust alternative to the traditional estimators and demonstrates the necessity and the merit of using the FSBP in finite-sample real practice.

翻译：由Rousseeuw和Hubert (1999) (RH99)提出的回归深度(RD)所诱导的最大深度估计量(又称深度中位数) ($\bs{\beta}^*_{RD}$)是回归分析中最流行的估计量之一。它具备与单变量位置估计量相似的卓越稳健性。事实上，$\bs{\beta}^*_{RD}$ 在渐近意义上可以抵抗高达 $33\%$ 的污染而不崩溃，而传统估计量(最小二乘和最小绝对偏差)的崩溃点为 $0\%$ (参见Van Aelst和Rousseeuw, 2000) (VAR00))。VAR00的结果具有开创性，但仅限于回归对称总体(具有严格正密度)以及 $\epsilon$-污染和最大偏差模型。在固定的有限样本实践中，评估估计量稳健性最常用的指标是有限样本崩溃点(FSBP) (Donoho和Huber (1983))。尽管文献中已有诸多尝试，但关于 $\bs{\beta}^*_{RD}$ 的FSBP仅获得了零星的局部结果，而其精确的FSBP在过去二十多年中一直悬而未决。此外，渐近崩溃值 $1/3$ (有限样本崩溃值递增序列的极限)在有限样本实践中是否相关？(或者说有限样本崩溃值与极限崩溃值之间的差异是什么？)这些问题在文献中尚未得到讨论。本文解决了上述问题，揭示了 $\bs{\beta}^*_{RD}$ 的回归深度与新获得的精确FSBP之间的内在联系。这论证了 $\bs{\beta}^*_{RD}$ 作为传统估计量的稳健替代方案的适用性，并展示了在有限样本实际应用中使用FSBP的必要性与价值。