Finite sample breakdown point of multivariate regression depth median

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.