Asymptotics for the maximum regression depth estimator
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Notions of depth in regression have been introduced and studied in the literature. Regression depth (RD) of Rousseeuw and Hubert (1999) (RH99), the most famous one, is a direct extension of Tukey location depth (Tukey (1975)) to regression. RD of RH99 satisfies all the desirable axiomatic properties in Zuo (2018) and therefore could serve as a real depth notion in regression. Like its location counterpart, the most remarkable advantage of notion of the depth in regression is to introduce directly the maximum (or deepest) regression depth estimator for regression parameters in a multi-dimensional setting. Classical questions for the maximum regression depth estimator include (i) is it a consistent estimator (or rather under what sufficient conditions, it is consistent) and (ii) is there any limiting distribution? Bai and He (1999) (BH99) intended to answer these questions. Unfortunately, they treated a closely related but different depth notion than the intended one. So the above questions remain open. Answering these questions is the main goal of this article.
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