Advertisement

Statistical Papers

, Volume 60, Issue 3, pp 641–666 | Cite as

Gini covariance matrix and its affine equivariant version

  • Xin DangEmail author
  • Hailin Sang
  • Lauren Weatherall
Regular Article
  • 168 Downloads

Abstract

We propose a new covariance matrix called Gini covariance matrix (GCM), which is a natural generalization of univariate Gini mean difference (GMD) to the multivariate case. The extension is based on the covariance representation of GMD by applying the multivariate spatial rank function. We study properties of GCM, especially in the elliptical distribution family. In order to gain the affine equivariance property for GCM, we utilize the transformation–retransformation (TR) technique and obtain an affine equivariant version GCM that turns out to be a symmetrized M-functional. The influence function of those two GCM’s are obtained and their estimation has been presented. Asymptotic results of estimators have been established. A closely related scatter Kotz functional and its estimator are also explored. Finally, asymptotical efficiency and finite sample efficiency of the TR version GCM are compared with those of sample covariance matrix, Tyler-M estimator and other scatter estimators under different distributions.

Keywords

Affine equivariance Efficiency Gini mean difference Influence function Scatter M-estimator Spatial rank Symmetrization 

Mathematics Subject Classification

62H10 62H12 

References

  1. Arslan O (2010) An alternative multivariate skew Laplace distribution: properties and estimation. Stat Pap 51:865–887MathSciNetCrossRefzbMATHGoogle Scholar
  2. Azzalini A, Genz A (2016) The R package ‘mnormt’: the multivariate normal and ‘t’ distributions (version 1.5-4). http://azzalini.stat.unipd.it/SW/Pkg-mnormt
  3. Carcea M, Serfling R (2015) A Gini autocovariance function for time series modeling. J Time Ser Anal 36:817–838CrossRefzbMATHGoogle Scholar
  4. Chakraborty B, Chaudhuri P (1996) On a transformation and re-transformation technique for constructing an affine equivariant multivariate median. Proc Am Math Soc 124(8):2539–2547MathSciNetCrossRefzbMATHGoogle Scholar
  5. Croux C, Ollila E, Oja H (2002) Sign and rank covariance matrices: statistical properties and application to principal components analysis. In: Dodge Y (ed) Statistical data analysis based on the L1-norm and related methods. Birkhauser, Basel, pp 257–271CrossRefGoogle Scholar
  6. Dümbgen L (1998) On Tyler’s M-functional of scatter in high dimension. Ann Inst Stat Math 50:471–491MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dümbgen L, Nordhausen K, Schuhmacher H (2014) fastM: fast computation of multivariate M-estimators. R package version 0.0-2. https://CRAN.R-project.org/package=fastM
  8. Dümbgen L, Pauly M, Schweizer T (2015) M-functionals of multivariate scatter. Stat Surv 9:32–105MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dümbgen L, Nordhausen K, Schuhmacher H (2016) New algorithms for M-estimation of multivariate scatter and location. J Multivar Anal 144:200–217MathSciNetCrossRefzbMATHGoogle Scholar
  10. Fang KT, Anderson TW (1990) Statistical inference in elliptically contoured and related distributions. Allerton Press, New YorkzbMATHGoogle Scholar
  11. Gerstenberger C, Vogel D (2015) On the efficiency of Gini’s mean difference. Stat Methods Appl 24(4):569–596MathSciNetCrossRefzbMATHGoogle Scholar
  12. Gini C (1914) Reprinted: on the measurement of concentration and variability of characters (2005). Metron LXIII(1):3–38MathSciNetzbMATHGoogle Scholar
  13. Hampel FR (1974) The influence curve and its role in robust estimation. J Am Stat Assoc 69:383–393MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WJ (1986) Robust statistics: the approach based on influence functions. Wiley, New YorkzbMATHGoogle Scholar
  15. Huber PJ (1967) The behavior of maximum likelihood estimates under nonstandard conditions. Proc Fifth Berkeley Symp Math Stat Probab 1:221–233MathSciNetzbMATHGoogle Scholar
  16. Huber PJ, Ronchetti EM (2009) Robust statistics, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  17. Hyvärinen A, Karhunen J, Oja E (2001) Independent component analysis. Wiley, New YorkCrossRefGoogle Scholar
  18. Koltchinskii VI (1997) M-estimation, convexity and quantiles. Ann Stat 25:435–477MathSciNetCrossRefzbMATHGoogle Scholar
  19. Koshevoy G, Mosler K (1997) Multivariate Gini indices. J Multivar Anal 60:252–276MathSciNetCrossRefzbMATHGoogle Scholar
  20. Koshevoy G, Möttönen J, Oja H (2003) Scatter matrix estimate based on the zonotope. Ann Stat 31:1439–1459MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kotz S (1975) Multivariate distributions at a cross-road. In: Patil GP, Kotz S, Ord JK (eds) Statistical distributions in scientific work, vol 1. Reidel Publication Company, DordrechtGoogle Scholar
  22. Maronna RA (1976) Robust M-estimators of multivariate location and scatter. Ann Stat 4:51–67MathSciNetCrossRefzbMATHGoogle Scholar
  23. Möttönen J, Oja H, Tienari J (1997) On the efficiency of multivariate spatial sign and rank tests. Ann Stat 25:542–552MathSciNetCrossRefzbMATHGoogle Scholar
  24. Nadarajah S (2003) The Kotz-type distribution with applications. Statistics 37:341–358MathSciNetCrossRefzbMATHGoogle Scholar
  25. Nair U (1936) The standard error of Gini’s mean difference. Biometrika 28:428–436CrossRefzbMATHGoogle Scholar
  26. Nordhausen K, Oja H (2011) Scatter matrices with independent block property and ISA. In: Proceedings of the 19th European signal processing conference (EUSIPCO 2011)Google Scholar
  27. Nordhausen K, Tyler DE (2015) A cautionary note on robust covariance plug-in methods. Biometrika 102:573–588MathSciNetCrossRefzbMATHGoogle Scholar
  28. Nordhausen K, Sirkiä S, Oja H, Tyler DE (2015) ICSNP: tools for multivariate nonparametrics. R package version 1.1-0. https://CRAN.R-project.org/package=ICSNP
  29. Oja H (1983) Descriptive statistics for multivariate distributions. Stat Probab Lett 1:327–332MathSciNetCrossRefzbMATHGoogle Scholar
  30. Oja H (2010) Multivariate nonparametric methods with R: an approach based on spatial signs and ranks. Springer, New YorkCrossRefzbMATHGoogle Scholar
  31. Oja H, Sirkiä S, Eriksson J (2006) Scatter matrices and independent component analysis. Austrian J Stat 35:175–189Google Scholar
  32. Ollila E, Oja H, Croux C (2003) The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies. J Multivar Anal 87:328–355MathSciNetCrossRefzbMATHGoogle Scholar
  33. Ollila E, Croux C, Oja H (2004) Influence function and asymptotic efficiency of the affine equivariant rank covariance matrix. Stat Sin 14:297–316MathSciNetzbMATHGoogle Scholar
  34. Paindaveine D (2008) A canonical definition of shape. Stat Probab Lett 78:2240–2247MathSciNetCrossRefzbMATHGoogle Scholar
  35. Roelant E, Van Aelst S (2007) An \(L_1\)-type estimator of multivariate location and shape. Stat Methods Appl 15:381–393CrossRefzbMATHGoogle Scholar
  36. Rousseeuw PJ, Croux C (1993) Alternatives to the median absolute deviation. J Am Stat Assoc 88:1273–1283MathSciNetCrossRefzbMATHGoogle Scholar
  37. Serfling R (1980) Approximation theorems of mathematical statistics. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  38. Serfling R (2010) Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardization. J Nonparametr Stat 22:915–936MathSciNetCrossRefzbMATHGoogle Scholar
  39. Serfling R, Xiao P (2007) A contribution to multivariate L-moments: L-comoment matrices. J Multivar Anal 98:1765–1781MathSciNetCrossRefzbMATHGoogle Scholar
  40. Sirkiä S, Taskinen S, Oja H (2007) Symmetrised M-estimators of multivariate scatter. J Multivar Anal 98:1611–1629MathSciNetCrossRefzbMATHGoogle Scholar
  41. Stamatis C, Steel H, Gordon S (1981) On the theory of elliptically contoured distributions. J Multivar Anal 11:368–385MathSciNetCrossRefzbMATHGoogle Scholar
  42. Taskinen S, Koch I, Oja H (2012) Robustifying principal component analysis with spatial sign vectors. Stat Probab Lett 82:765–774MathSciNetCrossRefzbMATHGoogle Scholar
  43. Tyler D (1987) A distribution-free M-estimator of multivariate scatter. Ann Stat 15:234–251MathSciNetCrossRefzbMATHGoogle Scholar
  44. Tyler D, Critchley F, Dümbgen L, Oja H (2009) Invariant coordinate selection. J R Stat Soc B 71:549–592MathSciNetCrossRefzbMATHGoogle Scholar
  45. Visuri S, Koivunen V, Oja H (2000) Sign and rank covariance matrices. J Stat Plan Inference 91:557–575MathSciNetCrossRefzbMATHGoogle Scholar
  46. Wang J (2009) A family of kurtosis orderings for multivariate distributions. J Multivar Anal 100:509–517MathSciNetCrossRefzbMATHGoogle Scholar
  47. Yitzhaki S (2003) Gini’s mean difference: a superior measure of variability for non-normal distribution. Metron Int J Stat 61:285–316MathSciNetzbMATHGoogle Scholar
  48. Yitzhaki S, Schechtman E (2013) The Gini methodology—a primer on a statistical methodology. Springer, New YorkCrossRefzbMATHGoogle Scholar
  49. Yu K, Dang X, Chen Y (2015) Robustness of the affine equivariant scatter estimator based on the spatial rank covariance matrix. Commun Stat Theory Methods 44:914–932MathSciNetCrossRefzbMATHGoogle Scholar
  50. Zografos K (2008) On Mardia’s and Song’s measures of kurtosis in elliptical distributions. J Multivar Anal 99:858–879MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MississippiUniversityUSA
  2. 2.BlueCross & BlueShield of MississippiFlowoodUSA

Personalised recommendations