On a transformation and re-transformation technique for constructing an affine

equivariant multivariate median

Authors:
Biman Chakraborty and Probal Chaudhuri

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

MSC (1991):
Primary 62A05, 62H12; Secondary 62E20

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

MathSciNet review:
1363452

Full-text PDF Free Access

Abstract | References | Similar Articles | 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 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.

**1.**Abdous, B. and Theodorescu, R. (1992), Note on the spatial quantile of a random vector.*Statistics & Probability Letters*,**13**, 333--336. MR**93d:62094****2.**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. 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**, 15--23. MR**90e:62078****4.**Bahadur, R.R. (1966), A note on quantiles in large samples.*The Annals of Mathematical Statistics*,**37**, 577--580. MR**32:6522****5.**Barnett, V. (1976), The ordering of multivariate data (with discussion).*Journal of the Royal Statistical Society, Series A*,**139**, 318--354. 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**, 1079--1090. MR**29:2904****7.**Brown, B. M. (1983), Statistical use of spatial median.*Journal of the Royal Statistical Society, Series B*,**45**, 25--30. MR**85a:62073****8.**Chaudhuri, P. (1992), Multivariate location estimation using extension of -estimates through -statistics type approach.*The Annals of Statistics*,**20**, 897--916. 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**, 1363--1370. MR**94j:62099****10.**Gower, J. C. (1974), The mediancenter.*Journal of the Royal Statistical Society, Series C*,**23**, 466--470.**11.**Haldane, J. B. S. (1948), Note on the median of a multivariate distribution.*Biometrika*,**35**, 414--415.**12.**Lancaster, H. O. (1969),*The Chi-Squared 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**, 405--414. MR**91d:62068****14.**Oja, H. (1983), Descriptive statistics for multivariate distributions.*Statistics & Probability Letters*,**1**, 327--332. MR**85a:62091****15.**Rao, C. R. (1988), Methodology based on the -norm in statistical inference.*Sankhya, Series A*,**50**, 289--313. MR**91e:62181****16.**Small, C. G. (1990), A survey of multidimensional medians.*International Statistical Review*,**58**, 263--277.**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. 523--531. MR**55:26**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
62A05,
62H12,
62E20

Retrieve articles in all journals with MSC (1991): 62A05, 62H12, 62E20

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

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