Finite sample breakdown point of multivariate regression depth median

Depth induced multivariate medians (multi-dimensional maximum depth estimators) in regression serve as robust alternatives to the traditional least squares and least absolute deviations estimators. The induced median ($\bs{\beta}^*_{RD}$) from regression depth (RD) of Rousseeuw and Hubert (1999) (RH99) is one of the most prevailing estimators in regression. The maximum regression depth median possesses outstanding robustness similar to the univariate location counterpart. Indeed, the %maximum depth estimator induced from $\mbox{RD}$, $\bs{\beta}^*_{RD}$ can, asymptotically, resist up to $33\%$ contamination without breakdown, in contrast to the $0\%$ for the traditional estimators %(i.e. they could break down by a single bad point) (see Van Aelst and Rousseeuw, 2000) (VAR00). The results from VAR00 are pioneering and innovative, yet they are limited to regression symmetric populations and the $\epsilon$-contamination and maximum bias model. With finite fixed sample size practice, the most prevailing measure of robustness for estimators is the finite sample breakdown point (FSBP) (Donoho (1982), Donoho and Huber (1983)). A lower bound (LB) of the FSBP for the $\bs{\beta}^*_{RD}$, which is not sharp, was given in RH99 (in a corollary of a conjecture). An exact FSBP (or even a sharper LB) for the $\bs{\beta}^*_{RD}$ remained open in the last two decades. This article establishes a sharper lower and upper bounds of (and an exact) FSBP for the $\bs{\beta}^*_{RD}$, revealing an intrinsic connection between the regression depth of $\bs{\beta}^*_{RD}$ and its FSBP. This justifies the employment of the $\bs{\beta}^*_{RD}$ as a robust alternative to the traditional estimators and demonstrating the necessity and the merit of using the FSBP in finite sample real practice instead of an asymptotic breakdown value.