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Some intriguing properties of Tukey’s half-space depth. (English) Zbl 1229.62063

Summary: For multivariate data, J. W. Tukey’s [Proc. Int. Congr. Math., Vancouver 1974, Vol. 2, 523–531 (1975; Zbl 0347.62002)] half-space depth is one of the most popular depth functions available in the literature. It is conceptually simple and satisfies several desirable properties of depth functions. The Tukey median, the multivariate median associated with the half-space depth, is also a well-known measure of center for multivariate data with several interesting properties.
We derive and investigate some interesting properties of half-space depth and its associated multivariate median. These properties, some of which are counterintuitive, have important statistical consequences in multivariate analysis. We also investigate a natural extension of Tukey’s half-space depth and the related median for probability distributions on any Banach space (which may be finite- or infinite-dimensional) and prove some results that demonstrate anomalous behavior of half-space depth in infinite-dimensional spaces.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62A09 Graphical methods in statistics
46N30 Applications of functional analysis in probability theory and statistics

Citations:

Zbl 0347.62002
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References:

[1] Ajne, B. (1968). A simple test for uniformity of a circular distribution. Biometrika 55 343-354. · Zbl 0157.48502
[2] Chaudhuri, P. and Sengupta, D. (1993). Sign tests in multi-dimension: Inference based on the geometry of the data cloud. J. Amer. Statist. Assoc. 88 1363-1370. · Zbl 0792.62047
[3] Chow, Y.S. and Teicher, H. (2005). Probability Theory: Independence, Interchangeability, Martingales . New York: Springer. · Zbl 1049.60001
[4] Cuesta-Albertosa, J.A. and Nieto-Reyes, A. (2008). The Tukey and the random Tukey depths characterize discrete distributions. J. Multivariate Anal. 10 2304-2311. · Zbl 1274.62351
[5] Dang, X. and Serfling, R. (2010). Nonparametric depth-based multivariate outlier identifiers, and masking robustness properties. J. Statist. Plann. Inference 140 198-213. · Zbl 1191.62084
[6] Donoho, D. and Gasko, M. (1992). Breakdown properties of location estimates based half-space depth and projected outlyingness. Ann. Statist. 20 1803-1827. · Zbl 0776.62031
[7] Ghosh, A.K. and Chaudhuri, P. (2005). On data depth and distribution free discriminant analysis using separating surfaces. Bernoulli 11 1-27. · Zbl 1059.62064
[8] Ghosh, A.K. and Chaudhuri, P. (2005). On maximum depth classifiers. Scand. J. Statist. 32 328-350. · Zbl 1089.62075
[9] Hassairi, A. and Regaieg, O. (2008). On the Tukey depth of a continuous probability distribution. Statist. Probab. Lett. 78 2308-2313. · Zbl 1274.62111
[10] Koshevoy, G.A. (2002). The Tukey’s depth characterizes the atomic measure. J. Multivariate Anal. 83 360-364. · Zbl 1028.62040
[11] Koshevoy, G.A. (2003). Lift-zonoid and multivariate depths. In Developments in Robust Statistics (Vorau, 2001) 194-202. Heidelberg: Physica. · Zbl 05280050
[12] Liu, R. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405-414. · Zbl 0701.62063
[13] Liu, R., Parelius, J. and Singh, K. (1999). Multivariate analysis of the data depth: Descriptive statistics and inference. Ann. Statist. 27 783-858. · Zbl 0984.62037
[14] Lopez-Pintado, S. and Romo, J. (2006). Depth based classification for functional data. In DIMACS Ser. Math. and Theo. Comp. Sci. (R. Liu and R. Serfling, Eds.) 72 103-119. Providence, RI: Amer. Math. Soc.
[15] Mizera, I. and Muller, C.H. (2004). Location-scale depth. J. Amer. Statist. Assoc. 99 949-966. · Zbl 1071.62032
[16] Mosler, K. (2002). Multivariate Dispersions, Central Regions and Depth . New York: Springer. · Zbl 1027.62033
[17] Nolan, D. (1992). Asymptotics for multivariate trimming. Stochastic Process. Appl. 42 157-169. · Zbl 0763.62007
[18] Serfling, R. (2006). Depth functions in nonparametric multivariate inference. In DIMACS Ser. Math. and Theo. Comp. Sci. (R. Liu and R. Serfling. Eds.) 72 1-16. Providence, RI: Amer. Math. Soc.
[19] Small, C.G. (1990). A survey of multidimensional medians. Inter. Statist. Rev. 58 263-277.
[20] Tukey, J. (1975). Mathematics and the picturing of data. In Proc. 1975 Inter. Cong. Math., Vancouver 523-531. Montreal: Canad. Math. Congress. · Zbl 0347.62002
[21] Vardi, Y. and Zhang, C.H. (2000). The multivariate L 1 -median and associated data depth. Proc. Natl. Acad. Sci. USA 97 1423-1426. · Zbl 1054.62067
[22] Zuo, Y. and Serfling, R. (2000). General notions of statistical depth function. Ann. Statist. 28 461-482. · Zbl 1106.62334
[23] Zuo, Y. (2003). Projection-based depth functions and associated medians. Ann. Statist. 31 1460-1490. · Zbl 1046.62056
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