Relationships between maximum depth and projection regression estimates

This article deals with the relationships between regression estimates based on projections (Ann. Statist. 21 (1993) 965) and on maximum depth (J. Amer. Statist. Assoc. 94 (1999) 388), which are shown to share the same basic idea. The maximum asymptotic bias of the latter is derived. A measure of residual smallness is defined, which turns out to be equivalent to depth. Two approximate algorithms based on subsampling are proposed to compute the maximum depth estimate: one based on the original definition, and the other on the new criterion. Finally, a Monte Carlo study is performed to compare the least median of squares estimate, the projection estimate, and the two approximate versions of the maximum depth estimate. It is seen that the latter have rather different behaviors, and the best overall performance corresponds to the projection estimate and one of the versions of the depth estimate. (C) 2001 Elsevier Science B.V. All rights reserved.