Computation of projection regression depth and its induced median
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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, exemplifies a direct extension of Tukey location depth (Tukey (1975)) to regression. The extension of another prevailing location depth, the projection depth (Liu (1992), and Zuo and Serfling (2000)), to regression is called the projection regression depth (PRD) (Zuo (2018a)). The computation issues of RD of RH99 have been discussed in RH99, Rousseeuw and Struyf (1998), and Liu and Zuo (2014). Those of PRD have never been dealt with. This article addresses the computation issues of PRD and its induced median (maximum depth estimator) in a regression setting, proposing exact algorithms for PRD with cost O(n^p+1) and approximate algorithms for PRD and its induced median with cost O(N_vn) and O(R(N_vN_βn+pN_β+N_vN_Itern)), respectively, where N_v is the total number of unit directions v tried, N_β is the total number of candidate regression parameters β tried, N_Iter is the total number of iterations carried out in an optimization algorithm, and R is the total number of replications. Examples and a simulation study reveal that the maximum depth estimator induced from PRD is favorable in terms of robustness and efficiency, compared with its major competitor, the maximum depth estimator induced from RD.
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