Depth weighted covariances

For years statisticans have sought ways to order/rank multidimensional observations. Statistical depth functions are proving to be a very promising method. They have become increasingly pursued as a general tool for multivariate and nonparametric data analysis and inference.

In this talk, the basic idea and some prevailing examples of depth functions are first discussed. General depth weighted means and covariances are then introduced and studied with an emphasis on "projection depth" weighted covariances, which include as a special case the Stahel-Donoho covariance estimator. Large sample behavior (including consistency and limiting distribution) and finite sample behavior (including breakdown and relative efficiency) of the projection depth weighted covariances are thoroughly examined. The influence function and the maximum bias of the projection depth weighted covariances are also discussed.

Unlike many high-breakdown competitors, the projection depth weighted covariances can integrate high breakdown point and high efficiency while enjoying a bounded-influence function and a moderate maximum bias curve. Comparisons with leading estimators on asymptotic relative efficiency and gross error sensitivity reveal that the projection depth weighted covariances behave very well overall and consequently represent very favorable choices of covariances.

REFERENCES:

(1) Y. Zuo, and R. Serfling (2000a). General notions of statistical depth function. The Annals of Statistics, 28(2), 461-482.
(2) Y. Zuo, and R. Serfling (2000b). Structural properties and convergence results for contours of sample statistical depth functions. The Annals of Statistics, 28(2), 483-499.
(3) Y. Zuo (2000). Projection based depth functions and associated medians. To appear in the Annals of Statistics.
(4) Y. Zuo, H. Cui, and X. He (2001). On the Stahel-Donoho estimator and depth-weighted means of multivariate datal. To appear in Annals of Statistics.
(5) Y. Zuo, H. Cui, and D. Young (2001). Influence function and Maximum bias of projection depth based estimators. Tentatively accepted by the Annals of Statistics.
(6) Y. Zuo and H. Cui (2002). Depth weighted covariances. Submitted to the Annals of Statistics.