  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.
