Computation of projection regression depth and its induced median

Abstract

Notions of depth in regression have been introduced and studied in the literature. The most famous example is Regression Depth (RD), which is a direct extension of location depth to regression. The projection regression depth (PRD) is the extension of another prevailing location depth, the projection depth, to regression. The computation issues of the RD have been discussed in the literature. The computation issues of the PRD have never been dealt with before. The computation issues of the PRD and its induced median (maximum depth estimator) in a regression setting are addressed now. For a given $\bs{\beta}\in\R^p$ exact algorithms for the PRD with cost $O(n^2\log n)$ ($p=2$) and $O(N(n, p)(p^{3}+n\log n+np^{1.5}+npN_{Iter}))$ ($p>2$) and approximate algorithms for the PRD and its induced median with cost respectively $O(N_{\mb{v}}np)$ and $O(Rp N_{\bs{\beta}}(p^2+nN_{\mb{v}}N_{Iter}))$ are proposed. Here $N(n, p)$ is a number defined based on the total number of $(p-1)$ dimensional hyperplanes formed by points induced from sample points and the $\bs{\beta}$; $N_{\mb{v}}$ is the total number of unit directions $\mb{v}$ utilized; $N_{\bs{\beta}}$ is the total number of candidate regression parameters $\bs{\beta}$ employed; $N_{Iter}$ is the total number of iterations carried out in an optimization algorithm; $R$ is the total number of replications. Furthermore, as the second major contribution, three PRD induced estimators, which can be computed up to 30 times faster than that of the PRD induced median while maintaining a similar level of accuracy are introduced. Examples and simulation studies reveal that the depth median induced from the PRD is favorable in terms of robustness and efficiency, compared to the maximum depth estimator induced from the RD, which is the current leading regression median.