Depth weighted scatter estimators. (English) Zbl 1065.62048

Summary: 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 \(\sqrt n\)-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.


62F35 Robustness and adaptive procedures (parametric inference)
62H12 Estimation in multivariate analysis
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
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[1] Chen, Z. and Tyler, D. E. (2002). The influence function and maximum bias of Tukey’s median. Ann. Statist. 30 1737–1759. · Zbl 1015.62058
[2] Cui, H. and Tian, Y. (1994). Estimation of the projection absolute median deviation and its application. J. Systems Sci. Math. Sci. 14 63–72. (In Chinese.) · Zbl 0826.62048
[3] Davies, P. L. (1987). Asymptotic behavior of \(S\)-estimates of multivariate location parameters and dispersion matrices. Ann. Statist. 15 1269–1292. JSTOR: · Zbl 0645.62057
[4] Donoho, D. L. (1982). Breakdown properties of multivariate location estimators. Ph.D. qualifying paper, Dept. Statistics, Harvard Univ.
[5] Donoho, D. L. and Huber, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann (P. J. Bickel, K. A. Doksum and J. L. Hodges, Jr., eds.) 157–184. Wadsworth, Belmont, CA. · Zbl 0523.62032
[6] Dümbgen, L. (1992). Limit theorem for the simplicial depth. Statist. Probab. Lett. 14 119–128. · Zbl 0758.60030
[7] Eaton, M. L. (1981). On the projections of isotropic distributions. Ann. Statist. 9 391–400. JSTOR: · Zbl 0463.62016
[8] Gather, U. and Hilker, T. (1997). A note on Tyler’s modification of the MAD for the Stahel–Donoho estimator. Ann. Statist. 25 2024–2026. · Zbl 0881.62033
[9] Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics: The Approach Based On Influence Functions . Wiley, New York. · Zbl 0593.62027
[10] He, X. and Simpson, D. G. (1993). Lower bounds for contamination bias: Globally minimax versus locally linear estimation. Ann. Statist. 21 314–337. JSTOR: · Zbl 0770.62023
[11] Huber, P. J. (1964). Robust estimation of a location parameter. Ann. Math. Statist. 35 73–101. · Zbl 0136.39805
[12] Kent, J. T. and Tyler, D. E. (1996). Constrained \(M\)-estimation multivariate location and scatter. Ann. Statist. 24 1346–1370. · Zbl 0862.62048
[13] Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405–414. JSTOR: · Zbl 0701.62063
[14] Liu, R. Y. (1992). Data depth and multivariate rank tests. In \(L_1\)-Statistical Analysis and Related Methods (Y. Dodge, ed.) 279–294. North-Holland, Amsterdam.
[15] Liu, R. Y., Parelius, J. M. and Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion). Ann. Statist. 27 783–858. · Zbl 0984.62037
[16] Lopuhaä, H. P. (1989). On the relation between \(S\)-estimators and \(M\)-estimators of multivariate location and covariance. Ann. Statist. 17 1662–1683. JSTOR: · Zbl 0702.62031
[17] Lopuhaä, H. P. (1999). Asymptotics of reweighted estimators of multivariate location and scatter. Ann. Statist. 27 1638–1665. · Zbl 0957.62017
[18] Maronna, R. A. (1976). Robust \(M\)-estimators of multivariate location and scatter. Ann. Statist. 4 51–67. JSTOR: · Zbl 0322.62054
[19] Maronna, R. A. and Yohai, V. J. (1995). The behavior of the Stahel–Donoho robust multivariate estimator. J. Amer. Statist. Assoc. 90 330–341. JSTOR: · Zbl 0820.62050
[20] Martin, R. D., Yohai, V. J. and Zamar, R. H. (1989). Min–max bias robust regression. Ann. Statist. 17 1608–1630. JSTOR: · Zbl 0713.62068
[21] Massé, J.-C. (2004). Asymptotics for the Tukey depth process, with an application to a multivariate trimmed mean. Bernoulli 10 397–419. · Zbl 1053.62021
[22] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Thoeory . Wiley, New York. · Zbl 0556.62028
[23] Pollard, D. (1984). Convergence of Stochastic Processes . Springer, New York. · Zbl 0544.60045
[24] Romanazzi, M. (2001). Influence function of halfspace depth. J. Multivariate Anal. 77 138–161. · Zbl 1033.62047
[25] Rousseeuw, P. J. (1985). Multivariate estimation with high breakdown point. In Mathematical Statistics and Applications (W. Grossmann, G. Pflug, I. Vincze and W. Wertz, eds.) 283–297. Reidel, Dordrecht. · Zbl 0609.62054
[26] Stahel, W. A. (1981). Breakdown of covariance estimators. Research Report 31, Fachgruppe für Statistik, ETH, Zürich.
[27] Tukey, J. W. (1975). Mathematics and the picturing of data. In Proc. International Congress of Mathematicians Vancouver 1974 2 523–531. Canadian Math. Congress, Montreal. · Zbl 0347.62002
[28] Tyler, D. E. (1982). Radial estimates and the test for sphericity. Biometrika 69 429–436. JSTOR: · Zbl 0501.62041
[29] Tyler, D. E. (1983). Robustness and efficiency properties of scatter matrices. Biometrika 70 411–420. JSTOR: · Zbl 0536.62042
[30] Tyler, D. E. (1994). Finite sample breakdown points of projection based multivariate location and scatter statistics. Ann. Statist. 22 1024–1044. JSTOR: · Zbl 0815.62015
[31] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics . Springer, New York. · Zbl 0862.60002
[32] Zuo, Y. (2003). Projection-based depth functions and associated medians. Ann. Statist. 31 1460–1490. · Zbl 1046.62056
[33] Zuo, Y. (2004). Robustness of weighted \(L_p\)-depth and \(L_p\)-median. Allg. Stat. Arch. 88 215–234. · Zbl 1294.62116
[34] Zuo, Y., Cui, H. and He, X. (2004). On the Stahel–Donoho estimator and depth-weighted means of multivariate data. Ann. Statist. 32 167–188. · Zbl 1105.62349
[35] Zuo, Y., Cui, H. and Young, D. (2004). Influence function and maximum bias of projection depth based estimators. Ann. Statist. 32 189–218. · Zbl 1105.62350
[36] Zuo, Y. and Serfling, R. (2000a). General notions of statistical depth function. Ann. Statist. 28 461–482. · Zbl 1106.62334
[37] Zuo, Y. and Serfling, R. (2000b). Structural properties and convergence results for contours of sample statistical depth functions. Ann. Statist. 28 483–499. · Zbl 1105.62343
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