On a transformation and re-transformation technique for constructing an affine equivariant multivariate median

Chakraborty, Biman; Chaudhuri, Probal

doi:10.1090/S0002-9939-96-03657-X

On a transformation and re-transformation technique for constructing an affine equivariant multivariate median
HTML articles powered by AMS MathViewer

by Biman Chakraborty and Probal Chaudhuri

Proc. Amer. Math. Soc. 124 (1996), 2539-2547

DOI: https://doi.org/10.1090/S0002-9939-96-03657-X

PDF | Request permission

Abstract:

An affine equivariant version of multivariate median is introduced. The proposed median is easy to compute and has some appealing geometric features that are related to the configuration of a multivariate data cloud. The transformation and re-transformation approach used in the construction of the median has some fundamental connection with the data driven co-ordinate system considered by Chaudhuri and Sengupta (1993, Journal of the American Statistical Association). Large sample statistical properties of the median are discussed and finite sample performance is investigated using Monte Carlo simulations.

References

B. Abdous and R. Theodorescu, Note on the spatial quantile of a random vector, Statist. Probab. Lett. 13 (1992), no. 4, 333–336. MR 1160756, DOI 10.1016/0167-7152(92)90043-5
Arcones, M.A., Chen, Z. and Giné, E. (1994), Estimators related to U-processes with applications to multivariate medians : Asymptotic normality. The Annals of Statistics, 22, 1460–1477.
G. Jogesh Babu and C. Radhakrishna Rao, Joint asymptotic distribution of marginal quantiles and quantile functions in samples from a multivariate population, J. Multivariate Anal. 27 (1988), no. 1, 15–23. MR 971169, DOI 10.1016/0047-259X(88)90112-1
R. R. Bahadur, A note on quantiles in large samples, Ann. Math. Statist. 37 (1966), 577–580. MR 189095, DOI 10.1214/aoms/1177699450
V. Barnett, The ordering of multivariate data, J. Roy. Statist. Soc. Ser. A 139 (1976), no. 3, 318–355. MR 445726, DOI 10.2307/2344839
Peter J. Bickel, On some alternative estimates for shift in the $p$-variate one sample problem, Ann. Math. Statist. 35 (1964), 1079–1090. MR 165624, DOI 10.1214/aoms/1177703266
B. M. Brown, Statistical uses of the spatial median, J. Roy. Statist. Soc. Ser. B 45 (1983), no. 1, 25–30. MR 701072, DOI 10.1111/j.2517-6161.1983.tb01226.x
Probal Chaudhuri, Multivariate location estimation using extension of $R$-estimates through $U$-statistics type approach, Ann. Statist. 20 (1992), no. 2, 897–916. MR 1165598, DOI 10.1214/aos/1176348662
Probal Chaudhuri and Debapriya Sengupta, Sign tests in multidimension: inference based on the geometry of the data cloud, J. Amer. Statist. Assoc. 88 (1993), no. 424, 1363–1370. MR 1245371, DOI 10.1080/01621459.1993.10476419
Gower, J. C. (1974), The mediancenter. Journal of the Royal Statistical Society, Series C, 23, 466–470.
Haldane, J. B. S. (1948), Note on the median of a multivariate distribution. Biometrika, 35, 414–415.
H. O. Lancaster, The chi-squared distribution, John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0253452
Regina Y. Liu, On a notion of data depth based on random simplices, Ann. Statist. 18 (1990), no. 1, 405–414. MR 1041400, DOI 10.1214/aos/1176347507
Hannu Oja, Descriptive statistics for multivariate distributions, Statist. Probab. Lett. 1 (1983), no. 6, 327–332. MR 721446, DOI 10.1016/0167-7152(83)90054-8
C. Radhakrishna Rao, Methodology based on the $L_1$-norm, in statistical inference, Sankhyā Ser. A 50 (1988), no. 3, 289–313. MR 1065546
Small, C. G. (1990), A survey of multidimensional medians. International Statistical Review, 58, 263–277.
John W. Tukey, Mathematics and the picturing of data, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 523–531. MR 0426989