On a transformation and retransformation technique for constructing an affine equivariant multivariate median
Authors:
Biman Chakraborty and Probal Chaudhuri
Journal:
Proc. Amer. Math. Soc. 124 (1996), 25392547
MSC (1991):
Primary 62A05, 62H12; Secondary 62E20
MathSciNet review:
1363452
Fulltext PDF Free Access
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Additional Information
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 retransformation approach used in the construction of the median has some fundamental connection with the data driven coordinate 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.
 1.
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
(93d:62094), http://dx.doi.org/10.1016/01677152(92)900435
 2.
Arcones, M.A., Chen, Z. and Giné, E. (1994), Estimators related to Uprocesses with applications to multivariate medians : Asymptotic normality. The Annals of Statistics, 22, 14601477. CMP 95:06
 3.
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
(90e:62078), http://dx.doi.org/10.1016/0047259X(88)901121
 4.
R.
R. Bahadur, A note on quantiles in large samples, Ann. Math.
Statist. 37 (1966), 577–580. MR 0189095
(32 #6522)
 5.
V.
Barnett, The ordering of multivariate data, J. Roy. Statist.
Soc. Ser. A 139 (1976), no. 3, 318–355. With a
discussion by R. L. Plackett, K. V. Mardia, R. M. Loynes, A. Huitson, G. M.
Paddle, T. Lewis, G. A. Barnard, A. M. Walker, F. Downton, P. J. Green,
Maurice Kendall, A. Robinson, Allan Seheult and D. H. Young. MR 0445726
(56 #4060)
 6.
Peter
J. Bickel, On some alternative estimates for shift in the
𝑝variate one sample problem, Ann. Math. Statist.
35 (1964), 1079–1090. MR 0165624
(29 #2904)
 7.
B.
M. Brown, Statistical uses of the spatial median, J. Roy.
Statist. Soc. Ser. B 45 (1983), no. 1, 25–30.
MR 701072
(85a:62073)
 8.
Probal
Chaudhuri, Multivariate location estimation using extension of
𝑅estimates through 𝑈statistics type approach, Ann.
Statist. 20 (1992), no. 2, 897–916. MR 1165598
(93h:62072), http://dx.doi.org/10.1214/aos/1176348662
 9.
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
(94j:62099)
 10.
Gower, J. C. (1974), The mediancenter. Journal of the Royal Statistical Society, Series C, 23, 466470.
 11.
Haldane, J. B. S. (1948), Note on the median of a multivariate distribution. Biometrika, 35, 414415.
 12.
H.
O. Lancaster, The chisquared distribution, John Wiley &
Sons, Inc., New YorkLondonSydney, 1969. MR 0253452
(40 #6667)
 13.
Regina
Y. Liu, On a notion of data depth based on random simplices,
Ann. Statist. 18 (1990), no. 1, 405–414. MR 1041400
(91d:62068), http://dx.doi.org/10.1214/aos/1176347507
 14.
Hannu
Oja, Descriptive statistics for multivariate distributions,
Statist. Probab. Lett. 1 (1983), no. 6,
327–332. MR
721446 (85a:62091), http://dx.doi.org/10.1016/01677152(83)900548
 15.
C.
Radhakrishna Rao, Methodology based on the 𝐿₁norm,
in statistical inference, Sankhyā Ser. A 50
(1988), no. 3, 289–313. MR 1065546
(91e:62181)
 16.
Small, C. G. (1990), A survey of multidimensional medians. International Statistical Review, 58, 263277.
 17.
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
(55 #26)
 1.
 Abdous, B. and Theodorescu, R. (1992), Note on the spatial quantile of a random vector. Statistics & Probability Letters, 13, 333336. MR 93d:62094
 2.
 Arcones, M.A., Chen, Z. and Giné, E. (1994), Estimators related to Uprocesses with applications to multivariate medians : Asymptotic normality. The Annals of Statistics, 22, 14601477. CMP 95:06
 3.
 Babu, G. J. and Rao, C. R. (1988), Joint asymptotic distribution of ma rginal quantile functions in samples from a multivariate population. Journal of Multivariate Analysis, 27, 1523. MR 90e:62078
 4.
 Bahadur, R.R. (1966), A note on quantiles in large samples. The Annals of Mathematical Statistics, 37, 577580. MR 32:6522
 5.
 Barnett, V. (1976), The ordering of multivariate data (with discussion). Journal of the Royal Statistical Society, Series A, 139, 318354. MR 56:4060
 6.
 Bickel, P. J. (1964), On some alternative estimates of shift in the variate one sample problem. Annals of Mathematical Statistics, 35, 10791090. MR 29:2904
 7.
 Brown, B. M. (1983), Statistical use of spatial median. Journal of the Royal Statistical Society, Series B, 45, 2530. MR 85a:62073
 8.
 Chaudhuri, P. (1992), Multivariate location estimation using extension of estimates through statistics type approach. The Annals of Statistics, 20, 897916. MR 93h:62072
 9.
 Chaudhuri, P. and Sengupta, D. (1993), Sign tests in multidimension : inference based on the geometry of the data cloud. Journal of the American Statistical Association, 88, 13631370. MR 94j:62099
 10.
 Gower, J. C. (1974), The mediancenter. Journal of the Royal Statistical Society, Series C, 23, 466470.
 11.
 Haldane, J. B. S. (1948), Note on the median of a multivariate distribution. Biometrika, 35, 414415.
 12.
 Lancaster, H. O. (1969), The ChiSquared Distribution. Wiley, New York. MR 40:6667
 13.
 Liu, R. Y. (1990), On a notion of data depth based on random simplices. The Annals of Statistics, 18, 405414. MR 91d:62068
 14.
 Oja, H. (1983), Descriptive statistics for multivariate distributions. Statistics & Probability Letters, 1, 327332. MR 85a:62091
 15.
 Rao, C. R. (1988), Methodology based on the norm in statistical inference. Sankhya, Series A, 50, 289313. MR 91e:62181
 16.
 Small, C. G. (1990), A survey of multidimensional medians. International Statistical Review, 58, 263277.
 17.
 Tukey, J. W. (1975), Mathematics and picturing data. In Proceedings of the International Congress of Mathematicians, Vancouver 1974 (Ed. R.D. James), vol. 2, pp. 523531. MR 55:26
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Additional Information
Biman Chakraborty
Affiliation:
Division of Theoretical Statistics & Mathematics, Indian Statistical Institute, 203 B. T. Road, Calcutta, 700035, India
Email:
res9421@isical.ernet.in
Probal Chaudhuri
Affiliation:
Division of Theoretical Statistics & Mathematics, Indian Statistical Institute, 203 B. T. Road, Calcutta, 700035, India
Email:
probal@isical.ernet.in
DOI:
http://dx.doi.org/10.1090/S000299399603657X
PII:
S 00029939(96)03657X
Keywords:
Affine transformation,
asymptotic distribution,
equivariance,
generalized variance
Received by editor(s):
November 18, 1994
Communicated by:
Wei Y. Loh
Article copyright:
© Copyright 1996
American Mathematical Society
