Open Access
Translator Disclaimer
October 2003 Projection-based depth functions and associated medians
Yijun Zuo
Ann. Statist. 31(5): 1460-1490 (October 2003). DOI: 10.1214/aos/1065705115


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.


Download Citation

Yijun Zuo. "Projection-based depth functions and associated medians." Ann. Statist. 31 (5) 1460 - 1490, October 2003.


Published: October 2003
First available in Project Euclid: 9 October 2003

zbMATH: 1046.62056
MathSciNet: MR2012822
Digital Object Identifier: 10.1214/aos/1065705115

Primary: 62F35 , 62H05
Secondary: 62G05 , 62H12

Keywords: asymptotic distribution , Breakdown point , consistency , depth contour , depth function , multivariate median , projection pursuit method , relative efficiency , robustness

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 5 • October 2003
Back to Top