Depth weighted scatter estimators

General depth weighted scatter estimators are introduced and investigated. For general depth functions, we find out that these affine equivariant scatter estimators are Fisher consistent and unbiased for a wide range of multivariate distributions, and show that the sample scatter estimators are strong and √n-consistent and asymptotically normal, and the influence functions of the estimators exist and are bounded in general. We then concentrate on a specific case of the general depth weighted scatter estimators, the projection depth weighted scatter estimators, which include as a special case the well-known Stahel-Donoho scatter estimator whose limiting distribution has long been open until this paper. Large sample behavior, including consistency and asymptotic normality, and efficiency and finite sample behavior, including breakdown point and relative efficiency of the sample projection depth weighted scatter estimators, are thoroughly investigated. The influence function and the maximum bias of the projection depth weighted scatter estimators are derived and examined. Unlike typical high-breakdown competitors, the projection depth weighted scatter estimators 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 scatter estimators behave very well overall and, consequently, represent very favorable choices of affine equivariant multivariate scatter estimators.

[1]  John Law,et al.  Robust Statistics—The Approach Based on Influence Functions , 1986 .

[2]  J. Massé.  Asymptotics for the Tukey depth process, with an application to a multivariate trimmed mean , 2004 .

[3]  Robb J. Muirhead,et al.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[4]  Y. Zuo.  Robustness of weighted Lp–depth and Lp–median , 2004 .

[5]  David E. Tyler.  Finite Sample Breakdown Points of Projection Based Multivariate Location and Scatter Statistics , 1994 .

[6]  H. P. Lopuhaä.  On the relation between S-estimators and M-estimators of multivariate location and covariance , 1989 .

[7]  V. Yohai,et al.  Min-Max Bias Robust Regression. , 1989 .

[8]  Regina Y. Liu.  On a Notion of Data Depth Based on Random Simplices , 1990 .

[9]  Victor J. Yohai,et al.  The Behavior of the Stahel-Donoho Robust Multivariate Estimator , 1995 .

[10]  Frederick R. Forst,et al.  On robust estimation of the location parameter , 1980 .

[11]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[12]  David E. Tyler,et al.  The influence function and maximum bias of Tukey's median , 2002 .

[13]  P. L. Davies,et al.  Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices , 1987 .

[14]  J. Tukey.  Mathematics and the Picturing of Data , 1975 .

[15]  L. Dümbgen.  Limit theorems for the simplicial depth , 1992 .

[16]  Xuming He,et al.  On the Stahel-Donoho estimator and depth-weighted means of multivariate data , 2003 .

[17]  D. Pollard.  Convergence of stochastic processes , 1984 .

[18]  R. Serfling,et al.  General notions of statistical depth function , 2000 .

[19]  H Cui.  ESTIMATION OF PROJECTION ABSOLUTE MEDIANDEVIATION AND ITS APPLICATION , 1994 .

[20]  P. Rousseeuw.  Multivariate estimation with high breakdown point , 1985 .

[21]  M. Romanazzi.  Influence Function of Halfspace Depth , 2001 .

[22]  Regina Y. Liu,et al.  A Quality Index Based on Data Depth and Multivariate Rank Tests , 1993 .

[23]  Hengjian Cui,et al.  Influence function and maximum bias of projection depth based estimators , 2003 .

[24]  David E. Tyler.  Robustness and efficiency properties of scatter matrices , 1983 .

[25]  Asymptotics for the Tukey Depth , .

[26]  D. G. Simpson,et al.  Lower Bounds for Contamination Bias: Globally Minimax Versus Locally Linear Estimation , 1993 .

[27]  Ursula Gather,et al.  A note on Tyler's modification of the MAD for the Stahel-Donoho estimator , 1997 .

[28]  Regina Y. Liu,et al.  Multivariate analysis by data depth: descriptive statistics, graphics and inference, (with discussion and a rejoinder by Liu and Singh) , 1999 .

[29]  David E. Tyler.  Radial estimates and the test for sphericity , 1982 .

[30]  Morris L. Eaton,et al.  On the Projections of Isotropic Distributions , 1981 .

[31]  R. Maronna.  Robust $M$-Estimators of Multivariate Location and Scatter , 1976 .

[32]  H. P. Lopuhaä.  ASYMPTOTICS OF REWEIGHTED ESTIMATORS OF MULTIVARIATE LOCATION AND SCATTER , 1999 .

[33]  Y. Zuo.  Projection-based depth functions and associated medians , 2003 .