On general notions of depth for regression
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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) and projection depth (Stahel (1981) and Donoho (1982), Liu (1992), Zuo and Serfling (2000) 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 Marrona and Yohai (1993) (MY93) and Carrizosa (1996). Do these depth notions possess desirable properties? Whether they can really serve as depth functions in regression remains an open question since unlike in the location case, there is no set of criteria to evaluate regression depth notions. Proposing evaluating criteria for regression depth notions and examining the existing ones with respect to the gauges is the main goal of this article. Extending the eminent projection depth in location to regression is a minor objective of the article. As an application or by-product of depth notion in regression, a general approach based on the min-max stratagem for estimating regression parameters is advanced. It is the median-type depth induced deepest estimating functionals for regression parameters and manifests one of the prominent advantages of notions of depth. Investigating and verifying the robustness of deepest regression depth estimating functionals is another major goal of this article. The asymptotic properties of depth functions, depth induced regions and deepest estimating functionals are examined in the article to achieve its third major goal.
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