Finite sample breakdown point of multivariate regression depth median
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Depth induced multivariate medians (multi-dimensional maximum depth estimators) in regression serve as robust alternatives to the traditional least squares and least absolute deviations estimators. The induced median (β^*_RD) from regression depth (RD) of Rousseeuw and Hubert (1999) (RH99) is one of the most prevailing estimators in regression. The maximum regression depth median possesses the outstanding robustness similar to the univariate location counterpart. Indeed, the estimator induced from , β^*_RD can, asymptotically, resist up to 33% contamination without breakdown, in contrast to the 0% for the traditional estimators point) (see Van Aelst and Rousseeuw, 2000) (VAR00). The results from VAR00 are pioneering and innovative, yet they are limited to regression symmetric populations and the ϵ-contamination and maximum bias model. With finite fixed sample size practice, the most prevailing measure of robustness for estimators is the finite sample breakdown point (FSBP) (Donoho (1982), Donoho and Huber (1983)). A lower bound of the FSBP for the β^*_RD was given in RH99 (in a corollary of a conjecture). This article establishes a sharper lower bound and an upper bound of the FSBP for the β^*_RD, revealing an intrinsic connection between the regression depth of β^*_RD and its FSBP, justifying the employment of the β^*_RD as a robust alternative to the traditional estimators and demonstrating the necessity and the merit of using FSBP in finite sample real practice instead of an asymptotic breakdown value.
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