论文标题
回归深度中位数的非反应鲁棒性分析
Non-asymptotic robustness analysis of regression depth median
论文作者
论文摘要
从Rousseuw和Hubert（1999）（RH99）的回归深度（RD）引起的最大深度估计器（又称深度中间）（$ \bsβ*_ {rd} $）是回归中最普遍的估计量之一。它具有与单变量位置对应物相似的出色鲁棒性。实际上，$ \bsβ^*_ {rd} $可以偶然地抵抗$ 33 \％$污染而不会崩溃的$ 33 \％$，与传统（最小二乘和最小值和绝对偏差）的$ 0 \％$相比（请参阅Van van aelst and Rousseeuw，2000）。 VAR00的结果是开创性的，但它们仅限于回归对称人群（严格的正密度）和$ε$ - 污染和最大偏置模型。通过固定的有限样本尺寸实践，估计器的鲁棒性最为普遍的方法是有限样本分解点（FSBP）（Donoho and Huber（1983））。尽管文献中进行了许多尝试，但仅获得了$ \bsβ^*_ {rd} $对FSBP的零星部分结果，而在过去的20年中，$ \bsβ^*_ {rd} $的精确FSBP仍被打开。此外，在有限样本实践中相关的渐近分解值$ 1/3 $（有限样本分解值增加的限制）？ （或有限样本和极限分解值有什么区别？）。这种讨论尚未在文献中进行。本文解决了上述问题，揭示了$ \bsβ^*_ {rd} $的回归深度与新获得的精确FSBP之间的内在联系。它证明了$ \bsβ^*_ {rd} $的利用合理性，是传统估计器的强大替代方案，并证明了在有限样本的实际实践中使用FSBP的必要性和优点。
The maximum depth estimator (aka depth median) ($\bsβ^*_{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, $\bsβ^*_{RD}$ 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 $ε$-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 $\bsβ^*_{RD}$ were obtained whereas an exact FSBP for $\bsβ^*_{RD}$ 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 $\bsβ^*_{RD}$ and the newly obtained exact FSBP. It justifies the employment of $\bsβ^*_{RD}$ as a robust alternative to the traditional estimators and demonstrates the necessity and the merit of using the FSBP in finite-sample real practice.
