Grübel, Rudolf Orthogonalization of multivariate location estimators: The orthomedian. (English) Zbl 0867.62048 Ann. Stat. 24, No. 4, 1457-1473 (1996). Summary: The coordinatewise median of a multivariate data set is a highly robust location estimator, but it depends on the choice of coordinates. A popular alternative which avoids this drawback is the spatial median, defined as the value that minimizes the sum of distances to the individual data points. In this paper we introduce and discuss another orthogonal equivariant version of the multivariate median, obtained by averaging the coordinatewise median over all orthogonal transformations. We investigate the asymptotic behavior of this estimator and compare it to the spatial median. Cited in 4 Documents MSC: 62H12 Estimation in multivariate analysis 62F35 Robustness and adaptive procedures (parametric inference) 62F12 Asymptotic properties of parametric estimators Keywords:orthogonal equivariance; asymptotic normality; coordinatewise median; robust location estimator; spatial median; multivariate median; averaging; orthogonal transformations Software:AS 143; AS 127 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] ANDERSON, T. W., INGRAM, I. and UNDERHILL, L. G. 1987. Generation of random orthogonal matrices. SIAM J. Sci. Statist. Comput. 8 625 629. Z. · Zbl 0637.65004 · doi:10.1137/0908055 [2] ARCONES, M. A. and GINE, E. 1992. On the bootstrap of M-estimators and other statistical Ź. functionals. In Exploring the Limits of Bootstrap R. LePage and L. Billard, eds. 13 47. Wiley, New York.Z. · Zbl 0842.62027 [3] BEDALL, F. K. and ZIMMERMANN, H. 1979. AS143: the mediancentre. J. Roy. Statist. Soc. Ser. C 28 325 328. Z. · Zbl 0464.65100 [4] BROWN, B. M. 1983. Statistical uses of the spatial median. J. Roy. Statist. Soc. Ser. B 45 25 30. Z. JSTOR: · Zbl 0508.62046 [5] FANG, K. T. and WANG, Y. 1994. Number-Theoretic Methods in Statistics. Chapman and Hall, London. Z. · Zbl 0955.62620 · doi:10.1214/ss/1177010392 [6] FELLER, W. 1971. An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York. Z. · Zbl 0219.60003 [7] GILL, R. D. 1989. Nonand semi-parametric maximum likelihood estimators and the von Mises Z. method Part I. Scand. J. Statist. 16 97 128. Z. · Zbl 0688.62026 [8] GRUBEL, R. 1988. The length of the shorth. Ann. Statist. 16 619 628. \" Z. · Zbl 0664.62040 · doi:10.1214/aos/1176350823 [9] GRUBEL, R. and ROCKE, D. M. 1990. On the cumulants of affine equivariant estimators in ëlliptical families. J. Multivariate Anal. 35 203 222. Z. · Zbl 0725.62050 · doi:10.1016/0047-259X(90)90025-D [10] HAMPEL, F. R., RONCHETTI, E. M., ROUSSEEUW, P. J. and STAHEL, W. A. 1986. Robust Statistics: The Approach Based on Influence Functions. Wiley, New York. Z. · Zbl 0593.62027 [11] HEIBERGER, R. M. 1978. Algorithm 127. Generation of random orthogonal matrices. Appl. Statist. 27 199 206. Z. · Zbl 0433.65006 · doi:10.2307/2346957 [12] LOPUHAA, H. P. and ROUSSEEUW, P. J. 1991. Breakdown points of affine equivariant estimators öf multivariate location and covariance matrices. Ann. Statist. 19 229 248. Z. · Zbl 0733.62058 · doi:10.1214/aos/1176347978 [13] POLLARD, D. 1984. Convergence of Stochastic Processes. Springer, New York. Z. · Zbl 0544.60045 [14] SMALL, C. G. 1990. A survey of multidimensional medians. Internat. Statist. Rev. 58 263 277. Z. [15] TANNER, M. A. and THISTED, R. A. 1982. A remark on AS 127. Generation of random orthogonal matrices. Appl. Statist. 31 190 192. Z. [16] VAN DER VAART, A. W. and WELLNER, J. A. 1996. Weak Convergence and Empirical Processes. Springer, New York. Z. · Zbl 0862.60002 [17] VERVAAT, W. 1972. Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrsch. Verw. Gebiete 23 245 253. · Zbl 0238.60018 · doi:10.1007/BF00532510 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.