  Depth notions in location have fascinated tremendous attention in the
literature. In fact data depth and its applications remain one of the most
active research topics in statistics in the last two decades. Most favored
notions of depth in location include Tukey (1975) halfspace depth (HD), Liu
(1990) simplicial depth, and projection depth (Stahel (1981) and Donoho (1982),
Liu (1992), Zuo and Serfling (2000) (ZS00) and Zuo (2003)), among others.
  Depth notions in regression have also been proposed, sporadically
nevertheless. Regression depth (RD) of Rousseeuw and Hubert (1999) (RH99) is
the most famous one which is a direct extension of Tukey HD to regression.
Others include Carrizosa (1996) and the ones induced from Marrona and Yohai
(1993) (MY93) proposed in this article. Is there any relationship between
Carrizosa depth and RD of RH99? Do these depth notions possess desirable
properties? What are the desirable properties? Can existing notions really
serve as depth functions in regression? These questions remain open.
  Revealing the equivalence between Carrizosa depth and RD of RH99; expanding
location depth evaluating criteria in ZS00 for regression depth notions;
examining the existing regression notions with respect to the gauges; and
proposing the regression counterpart of the eminent projection depth in
location are the four major objectives of the article.
