Effect of kurtosis on efficiency of some multivariate medians

References

• Arcones, M.A., Chen, Z., and Gine, E. (1994), ‘Estimators Related to U-Processes with Applications to Multivariate Medians: Asymptotic Normality’, The Annals of Statistics, 22, 1460–1477. doi: 10.1214/aos/1176325637
   Web of Science ®Google Scholar
• Arevalillo, J.M., and Navarro, H. (2012), ‘A Study of the Effect of Kurtosis on Discriminant Analysis Under Elliptical Populations’, Journal of Multivariate Analysis, 107, 53–63. doi: 10.1016/j.jmva.2012.01.011
   Web of Science ®Google Scholar
• Averous, J., and Meste, M. (1994), ‘Multivariate Kurtosis in -Sense’, Statistics and Probability Letters, 19, 281–284. doi: 10.1016/0167-7152(94)90177-5
   Web of Science ®Google Scholar
• Babu, G.J., and Rao, C.R. (1988), ‘Joint Asymptotic Distribution of Marginal Quantiles and Quantile Functions in Samples from a Multivariate Population’, Journal of Multivariate Analysis, 27, 15–23. doi: 10.1016/0047-259X(88)90112-1
   Web of Science ®Google Scholar
• Bickel, P.J. (1964), ‘On Some Alternative Estimates for Shift in the p-Variate One Sample Problem’, The Annals of Mathematical Statistics, 35, 1079–1090. doi: 10.1214/aoms/1177703266
   Google Scholar
• Bickel, P.J. (1965), ‘On Some Asymptotically Nonparametric Competitors of Hotelling's ’, The Annals of Mathematical Statistics, 36, 160–173. doi: 10.1214/aoms/1177700280
   Google Scholar
• Brown, B.M. (1983), ‘Statistical Uses of the Spatial Median’, Journal of the Royal Statistical Society, Series B, 45, 25–30.
   Google Scholar
• Chakraborty, B., and Chaudhuri, P. (1996), ‘On a Transformation and Re-Transformation Technique for Constructing an Affine Equivariant Multivariate Median’, Proceedings of the American Mathematical Society, 124, 2539–2547. doi: 10.1090/S0002-9939-96-03657-X
   Web of Science ®Google Scholar
• Chakraborty, B., Chaudhuri, P., and Oja, H. (1998), ‘Operating Transformation Retransformation on Spatial Median and Angle Test’, Statistica Sinica, 8, 767–784.
   Web of Science ®Google Scholar
• Chaudhuri, P. (1992), ‘Multivariate Location Estimation Using Extension of R-Estimates Through U-Statistics Type Approach’, The Annals of Statistics, 20, 897–916. doi: 10.1214/aos/1176348662
   Web of Science ®Google Scholar
• Dumbgen, L. (1998), ‘On Tyler's M-Functional of Scatter in High Dimension’, Annals of the Institute of Statistical Mathematics, 50, 471–491. doi: 10.1023/A:1003573311481
   Web of Science ®Google Scholar
• Fang, K.T., Kotz, S., and Ng, K.W. (1990), Symmetric Multivariate and Related Distributions, London: Chapman and Hall.
   Google Scholar
• Fritz, H., Filzmoser, P., and Croux, C. (2012), ‘A Comparison of Algorithms for the Multivariate -Median’, Computational Statistics, 27, 393–410. doi: 10.1007/s00180-011-0262-4
   Web of Science ®Google Scholar
• Hettmansperger, T.P., Nyblom, J., and Oja, H. (1994), ‘Affine Invariant Multivariate One-Sample Sign Tests’, Journal of the Royal Statistical Society, Series B, 56, 221–234.
   Google Scholar
• Hettmansperger, T.P., and Randles, R.H. (2002), ‘A Practical Affine Equivariant Multivariate Median’, Biometrika, 89, 851–860. doi: 10.1093/biomet/89.4.851
   Web of Science ®Google Scholar
• Lehmann, E.L., and Casella, G. (1998), Theory of Point Estimation, New York: Springer-Verlag.
   Google Scholar
• Liu, R.Y., Parelius, J.M., and Singh, K. (1999), ‘Multivariate Analysis by Data Depth: Descriptive Statistics, Graphics and Inference (with discussion)’, The Annals of Statistics, 27, 783–858.
   Web of Science ®Google Scholar
• Magyar, A., and Tyler, D.E. (2011), ‘The Asymptotic Efficiency of the Spatial Median for Elliptically Symmetric Distributions’, Sankhya B, 73, 165–192. doi: 10.1007/s13571-011-0032-x
   Google Scholar
• Mahalanobis, P.C. (1936), ‘On the Generalized Distance in Statistics’, Proceedings of the National Academy of Sciences, India, 12, 49–55.
   Google Scholar
• Malkovich, J.F., and Afifi, A.A. (1973), ‘On Tests for Multivariate Normality’, Journal of the American Statistical Association, 68, 176–179. doi: 10.1080/01621459.1973.10481358
   Web of Science ®Google Scholar
• Mardia, K.V. (1970), ‘Measures of Multivariate Skewness and Kurtosis with Applications’, Biometrika, 57, 519–530. doi: 10.1093/biomet/57.3.519
   Web of Science ®Google Scholar
• Merkle, M. (2001), ‘Conditions for Convexity of a Derivative and Applications to the Gamma and Digamma Function’, Facta Universitatis (NIS) Series: Mathematics and Informatics, 16, 13–20.
   Google Scholar
• Mood, A.M. (1941), ‘On the Joint Distribution of the Medians in Samples from a Multivariate Population’, The Annals of Mathematical Statistics, 12, 268–278. doi: 10.1214/aoms/1177731709
   Google Scholar
• Mottonen, J., Nordhausen, K., and Oja, H. (2010), ‘Asymptotic Theory of the Spatial Median’, IMS Collections, Nonparametrics and Robustness in Modern Statistical Inference and Time Series Analysis: A Festschrift in honor of Professor Jana Jureckova, 7, 182–193.
   Google Scholar
• Mottonen, J., Oja, H., and Tienari, J. (1997), ‘On the Efficiency of Multivariate Spatial Sign and Rank Tests’, The Annals of Statistics, 25, 542–552. doi: 10.1214/aos/1031833663
   Web of Science ®Google Scholar
• Niinimaa, A., Oja, H., and Tableman, M. (1990), ‘The Finite-Sample Breakdown Point of the Oja Bivariate Median and of the Corresponding Half-Samples Version’, Statistics and Probability Letters, 10, 325–328. doi: 10.1016/0167-7152(90)90050-H
   Web of Science ®Google Scholar
• Oja, H. (1983), ‘Descriptive Statistics for Multivariate Distributions’, Statistics and Probability Letters, 1, 327–332. doi: 10.1016/0167-7152(83)90054-8
   Google Scholar
• Oja, H. (1999), ‘Affine Invariant Multivariate Sign and Rank Tests and Corresponding Estimates: A Review’, Scandinavian Journal of Statistics, 26, 319–343. doi: 10.1111/1467-9469.00152
   Web of Science ®Google Scholar
• Oja, H., and Niinimma, A. (1985), ‘Asymptotic Properties of the Generalized Median in the Case of Multivariate Normality’, Journal of the Royal Statistical Society, Series B, 47, 372–377.
   Google Scholar
• Pollard, D. (1984), Convergence of Stochastic Processes, New York: Springer.
   Google Scholar
• Randles, R.H. (1989), ‘A Distribution-Free Multivariate Sign Test Based On Interdirections’, Journal of the American Statistical Association, 84, 1045–1050. doi: 10.1080/01621459.1989.10478870
   Web of Science ®Google Scholar
• Rao, C.R. (1988), ‘Methodology Based on the -Norm in Statistical Inference’, Sankhya, Series A, 50, 289–313.
   Google Scholar
• Serfling, R. (2004), ‘Nonparametric Multivariate Descriptive Measures Based on Spatial Quantiles’, Journal of Statistical Planning and Inference, 123, 259–278. doi: 10.1016/S0378-3758(03)00156-3
   Web of Science ®Google Scholar
• Small, C.G. (1990), ‘A Survey of Multidimensional Medians’, International Statistical Review, 58, 263–277. doi: 10.2307/1403809
   Web of Science ®Google Scholar
• Srivastava, M.S. (1984), ‘A Measure of Skewness and Kurtosis and a Graphical Method for Assessing Multivariate Normality’, Statistics and Probability Letters, 2, 263–267. doi: 10.1016/0167-7152(84)90062-2
   Web of Science ®Google Scholar
• Tyler, D.E. (1987), ‘A Distribution-Free M-Estimator of Multivariate Scatter’, The Annals of Statistics, 15, 234–251. doi: 10.1214/aos/1176350263
   Web of Science ®Google Scholar
• Tyler, D.E., Critchley, F., Dumbgen, L., and Oja, H. (2009), ‘Invariant Co-Ordinate Selection’, Journal of the Royal Statistical Society, Series B, 71, 549–592. doi: 10.1111/j.1467-9868.2009.00706.x
   Google Scholar
• van Zwet, W.R. (1964), Convex Transformations of Random Variables, Amsterdam: Mathematish Centrum.
   Google Scholar
• Wang, J. (2009), ‘A Family of Kurtosis Orderings for Multivariate Distributions’, Journal of Multivariate Analysis, 100, 509–517. doi: 10.1016/j.jmva.2008.06.001
   Web of Science ®Google Scholar
• Wang, J., and Serfling, R. (2005), ‘Nonparametric Multivariate Kurtosis and Tailweight Measures’, Journal of Nonparametric Statistics, 17, 441–456. doi: 10.1080/10485250500039130
   Web of Science ®Google Scholar
• Wang, J., and Zhou, W. (2012), ‘A Generalized Multivariate Kurtosis Ordering and Its Applications’, Journal of Multivariate Analysis, 107, 69–180. doi: 10.1016/j.jmva.2012.01.009
   Web of Science ®Google Scholar