On General Notions of Depth for Regression

Abstract

Depth notions in location have generated tremendous attention in the literature. In fact, data depth and its applications remain as one of the most active research topics in statistics over the last three decades. Most favored notions of depth in location include Tukey (In Proceedings of the International Congress of Mathematicians $($Vancouver, B.C., 1974$)$, Vol. 2 (1975) 523–531) half-space depth (HD), Liu (Ann. Statist. 18 (1990) 405–414) simplicial depth and projection depth (PD) (Stahel (1981) and Donoho (1982), Liu (In $L_{1}$-Statistical Analysis and Related Methods $($Neuchâtel, 1992$)$ (1992) 279–294 North-Holland), Zuo and Serfling (Ann. Statist. 28 (2000) 461–482) and (ZS00) and Zuo (Ann. Statist. 31 (2003) 1460–1490)), among others. Depth notions in regression have also been proposed sporadically, nevertheless. The regression depth (RD) of Rousseeuw and Hubert (J. Amer. Statist. Assoc. 94 (1999) 388–433) (RH99), the most famous, exemplifies a direct extension of Tukey HD to regression. Other notions include Carrizosa (J. Multivariate Anal. 58 (1996) 21–26) and the ones proposed in this article via modifying a functional in Maronna and Yohai (Ann. Statist. 21 (1993) 965–990) (MY93). Is there any relationship between Carrizosa depth and the RD of RH99? Do these depth notions possess desirable properties? What are the desirable properties? Can existing notions serve well as depth notions in regression? These questions remain open. The major objectives of the article include (i) revealing the connection between Carrizosa depth and RD of RH99; (ii) expanding location depth evaluating criteria in ZS00 for regression depth notions; (iii) examining the existing regression notions with respect to the gauges; and (iv) proposing the regression counterpart of the eminent location projection depth.