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Median balls: An extension of the interquantile intervals to multivariate distributions. (English) Zbl 0907.62061
For a probability distribution \(P\) on a Banach space \((X,\;\|.\|)\), interquantile intervals are defined by \([F^{-1} (\alpha), F^{-1} (1-\alpha)]\), and are used as central regions for studies of location, dispersion, skewness, or kurtosis. A median ball of \(P\) with nonnegative radius \(\lambda\) is any closed ball \(B(c^*,\lambda)\) with center \(c^*\), such that there exists a mapping \(\varphi^*\) in the set \(\Phi (c^*, \lambda)\) of measurable mappings from \(X\) to \(X\) for which \(\varphi^* (X) \subset B (c^*, \lambda)\), satisfying \(f(\varphi^*) =\inf_{c\in X} \inf_{\varphi\in \Phi(c, \lambda)}\). When \(\lambda =0\) they give the spatial median, and for a real distribution they are the interquantile intervals. Some properties are studied, and in particular it is shown that they possess robustness and equivariance properties similar to those of the spatial median.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60B11 Probability theory on linear topological spaces
60E05 Probability distributions: general theory
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[1] Abdous, B.; Theodorescu, R., Note on the spatial quantile of a random vector, Statist. probab. lett., 13, 333-336, (1992) · Zbl 0743.62040
[2] Averous, J.; Meste, M., Location, skewness and tailweight inL_s-sense: a coherent approach, Statistics, 21, 57-74, (1990) · Zbl 0764.62013
[3] Balanda, K.P.; MacGillivray, H.L., Kurtosis and spread, Canad. J. statist., 18, 17-30, (1990) · Zbl 0692.62010
[4] Bickel, P.J.; Lehmann, E.L., Descriptive statistics for nonparametric models (IV spread), Contributions to statistics, jaroslav Hajek memorial volume, (1979), Academia Prague, p. 33-40 · Zbl 0415.62015
[5] Breckling, J.; Chambers, R., M-quantiles, Biometrika, 75, 761-771, (1988) · Zbl 0653.62024
[6] D, Donoho, 1982, Breakdown Properties of Multivariate Location Estimators, Harvard University
[7] Ducharme, G.R.; Milasevic, P., Spatial Median and directional data, Biometrika, 74, 212-215, (1987) · Zbl 0607.62055
[8] Eaton, M.L.; Perlman, M.D., Concentration inequalities for multivariate distributions: I. multivariate normal distributions, Statist. probab. lett., 12, 487-504, (1991) · Zbl 0749.62036
[9] Einmahl, J.H.; Mason, D.M., Generalized quantile process, Ann. statist., 20, 1062-1078, (1992) · Zbl 0757.60012
[10] Giovagnoli, A.; Wynn, H.P., Multivariate dispersion orderings, Statist. probab. lett., 22, 325-332, (1995) · Zbl 0813.62048
[11] Grœneveld, R.A.; Meeden, G., Measuring skewness and kurtosis, The Statistician, 33, 391-399, (1984)
[12] Haberman, S.J., Concavity and estimation, Ann. statist., 17, 1631-1661, (1989) · Zbl 0699.62027
[13] Jensen, D.R., Invariant ordering and order preservation, Inequalities in statistics and probability, I.M.S. lecture notes, monographs series, 5, (1984), p. 26-34
[14] Johnson, N.L.; Kotz, S.; Balakrishnan, N., Continuous univariate distributions, (1995), Wiley New York · Zbl 0821.62001
[15] Kemperman, J.H.B., The Median of a finite measure on a Banach space, (), 217-230
[16] León, C.A.; Massé, J.C., A counterexample on the existence of theL1, Statist. probab. lett., 13, 117-120, (1992) · Zbl 0742.60007
[17] Liu, R.Y., On a notion of data depth based on random simplices, Ann. statist., 18, 405-414, (1990) · Zbl 0701.62063
[18] Liu, R.Y., Data depth and multivariate rank tests, (), 279-294
[19] Loewner, C., Über monotone matrixfunktionen, Math. Z., 38, 177-216, (1934) · Zbl 0008.11301
[20] MacGillivray, H.L.; Balanda, K.P., The relationships between skewness and kurtosis, Austr. J. statist., 30, 319-337, (1988) · Zbl 0707.62024
[21] Masse, J.-C.; Theodorescu, R., Halfplane trimming for bivariate distributions, J. multivariate anal., 48, 188-202, (1994) · Zbl 0790.60024
[22] Nolan, D., Asymptotics for multivariate trimming, Stochast. process. appl., 42, 157-169, (1992) · Zbl 0763.62007
[23] Oja, H., Descriptive statistics for multivariate distributions, Statist. probab. lett., 1, 327-332, (1983) · Zbl 0517.62051
[24] Rousseuw, P.J.; Leroy, A.M., Robust regression and outliers detection, (1987), Wiley New York
[25] Small, C.G., A survey of multidimensional medians, Intern. statist. rev., 58, 263-277, (1990)
[26] J. W. Tukey, 1975, Mathematics and the picturing of data, Proc. International Congress of Mathematicians, Vancouver, 2, 523, 531 · Zbl 0347.62002
[27] Valadier, M., La multi-application médianes conditionnelles, Wahrsch., 67, 279-282, (1984) · Zbl 0572.60013
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