Projection-based depth functions and associated medians

Abstract

A class of projection-based depth functions is introduced and studied. These projection-based depth functions possess desirable properties of statistical depth functions and their sample versions possess strong and order $\sqrt{n}$ uniform consistency. Depth regions and contours induced from projection-based depth functions are investigated. Structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained.

Affine equivariant multivariate medians induced from projection-based depth functions are probed. The limiting distributions as well as the strong and order $\sqrt{n}$ consistency of the sample projection medians are established. The finite sample performance of projection medians is compared with that of a leading depth-induced median, the Tukey halfspace median (induced from the Tukey halfspace depth function). It turns out that, with appropriate choices of univariate location and scale estimators, the projection medians have a very high finite sample breakdown point and relative efficiency, much higher than those of the halfspace median.

Based on the results obtained, it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators, respectively.