Advertisement

More good news on the HKM test for multivariate reflected symmetry about an unknown centre

  • Norbert HenzeEmail author
  • Celeste Mayer
Article
  • 3 Downloads

Abstract

We revisit the problem of testing for multivariate reflected symmetry about an unspecified point. Although this testing problem is invariant with respect to full-rank affine transformations, among the few hitherto proposed tests only a class of tests studied in Henze et al. (J Multivar Anal 87:275–297, 2003) that depends on a positive parameter a respects this property. We identify a measure of deviation \(\varDelta _a\) (say) from symmetry associated with the test statistic \(T_{n,a}\) (say), and we obtain the limit normal distribution of \(T_{n,a}\) as \(n \rightarrow \infty \) under a fixed alternative to symmetry. Since a consistent estimator of the variance of this limit normal distribution is available, we obtain an asymptotic confidence interval for \(\varDelta _a\). The test, when applied to a classical data set, strongly rejects the hypothesis of reflected symmetry, although other tests even do not object against the much stronger hypothesis of elliptical symmetry.

Keywords

Test for reflected symmetry Fixed alternatives Affine invariance Weighted \(L^2\)-statistic Elliptical symmetry 

Notes

Acknowledgements

The authors wish to thank two anonymous referees and a member of the Editorial Board for their careful reading of the manuscript and for many helpful suggestions.

References

  1. Aki, S. (1993). On nonparametric tests for symmetry in \({\mathbb{R}}^m\). Annals of the Institute of Statistical Mathematics, 45, 787–800.MathSciNetzbMATHGoogle Scholar
  2. Baringhaus, L., Henze, N. (1992). Limit distributions for Mardia’s measure of multivariate skewness. Annals of Statistics, 20, 1889–1902.MathSciNetzbMATHGoogle Scholar
  3. Baringhaus, L., Ebner, B., Henze, N. (2017). The limit distribution of weighted \(L^2\)-goodness-of-fit statistics under fixed alternatives, with applications. Annals of the Institute of Statistical Mathematics, 69, 969–995.MathSciNetzbMATHGoogle Scholar
  4. Batsidis, A., Martin, N., Pardo, L., Zografos, K. (2014). A necessary power divergence-type family of tests for testing elliptical symmetry. Journal of Statistical Computation and Simulation, 84, 57–83.MathSciNetGoogle Scholar
  5. Benjamini, Y., Yekutieli, D. (2001). The Control of the False Discovery Rate in Multiple Testing under Dependency. Annals of Statistics, 29, 1165–1188.MathSciNetzbMATHGoogle Scholar
  6. Bosq, D. (2000). Linear processes in function spaces. New York: Springer.zbMATHGoogle Scholar
  7. Bowman, A. W., Foster, P. J. (1993). Adaptive smoothing and density-based tests of multivariate normality. Journal of the American Statistical Association, 88, 529–537.MathSciNetzbMATHGoogle Scholar
  8. Dyckerhoff, R., Ley, C., Paindaveine, D. (2015). Depth-based run tests for bivariate central symmetry. Annals of the Institute of Statistical Mathematics, 67, 917–941.MathSciNetzbMATHGoogle Scholar
  9. Eaton, M. L., Perlman, M. D. (1973). The non-singularity of generalized sample covariance matrices. Annals of Statistics, 1, 710–717.MathSciNetzbMATHGoogle Scholar
  10. Einmahl, J. H. J., Gan, Z. (2016). Testing for central symmetry. Journal of Statistical Planning and Inference, 169, 27–33.MathSciNetzbMATHGoogle Scholar
  11. Gürtler, N. (2000). Asymptotic results on the BHEP tests for multivariate normality with fixed and variable smoothing parameter (in German). Doctoral dissertation. University of Karlsruhe.Google Scholar
  12. Heathcote, C. R., Rachev, S. T., Cheng, B. (1995). Testing multivariate symmetry. Journal of Multivariate Analysis, 54, 91–112.MathSciNetzbMATHGoogle Scholar
  13. Henze, N. (1997a). Limit laws for multivariate skewness in the sense of Móri, Rohatgi and Székely. Statistics & Probability Letters, 33, 299–307.MathSciNetzbMATHGoogle Scholar
  14. Henze, N. (1997b). Extreme smoothing and testing for multivariate normality. Statistics & Probability Letters, 35, 203–213.MathSciNetzbMATHGoogle Scholar
  15. Henze, N., Wagner, T. (1997). A new approach to the BHEP tests for multivariate normality. Journal of Multivariate Analysis, 62, 1–23.MathSciNetzbMATHGoogle Scholar
  16. Henze, N. (2002). Invariant tests for multivariate normality: a critical review. Statistical Papers, 43, 467–506.MathSciNetzbMATHGoogle Scholar
  17. Henze, N., Klar, B., Meintanis, S. (2003). Invariant tests for symmetry about an unspecified point based on the empirical characteristic function. Journal of Multivariate Analysis, 87, 275–297.MathSciNetzbMATHGoogle Scholar
  18. Janssen, A. (2000). Global power functions of goodness of fit tests. Annals of Statistics, 28, 239–253.MathSciNetzbMATHGoogle Scholar
  19. Ley, C., Verdebout, T. (2014). Simple optimal tests for circular reflective symmetry about a specified median direction. Statistica Sinica, 24, 1319–1339.MathSciNetzbMATHGoogle Scholar
  20. Madhava Rao, K. S., Raghunath, M. (2012). A simple nonparametric test for bivariate symmetry about a line. Journal of Statistical Planning and Inference, 142, 430–444.MathSciNetzbMATHGoogle Scholar
  21. Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57(3), 519–530.MathSciNetzbMATHGoogle Scholar
  22. Meintanis, S., Ngatchou-Wandji, J. (2012). Recent tests for symmetry with multivariate and structured data: a review. Nonparametric statistical methods and related topics, 35–73, World Scientific Publishing, Hackensack, NJ.Google Scholar
  23. Móri, T. F., Rohatgi, V. K., Székely, G. F. (1993). On multivariate skewness and kurtosis. Theory of Probability and its Applications, 38(1), 547–551.MathSciNetzbMATHGoogle Scholar
  24. Neuhaus, G., Zhu, L.-X. (1998). Permutation Tests for Reflected Symmetry. Journal of Multivariate Analysis, 67, 129–153.MathSciNetzbMATHGoogle Scholar
  25. Ngatchou-Wandji, J. (2009). Testing for symmetry in multivariate distributions. Statistical Methodology, 6, 230–250.MathSciNetzbMATHGoogle Scholar
  26. Partlett, C., Prakash, P. (2017). Measuring asymmetry and testing symmetry. Annals of the Institute of Statistical Mathematics, 96(2), 429–460.MathSciNetzbMATHGoogle Scholar
  27. Quessy, J.-F. (2016). On consistent nonparametric statistical tests of symmetry hypotheses. Symmetry 8, No.5, Art. 31, 19 pp.Google Scholar
  28. Royston, J. P. (1983). Some techniques for assessing multivariate normality based on the Shapiro-Wilk W. Applied Statistics, 32, 121–133.zbMATHGoogle Scholar
  29. Schott, J. R. (2002). Testing for elliptical symmetry in covariance-matrix-based analyses. Statistics & Probability Letters, 60, 395–404.MathSciNetzbMATHGoogle Scholar
  30. Serfling, R. (2006). Multivariate symmetry and asymmetry. In S. Kotz, N. Balakrishnan, C. B. Read & B. Vidacovic (Eds.), Encyclopedia of statistical sciences. 2nd ed., Vol. 8, pp. 5338–5345. New York: Wiley.Google Scholar
  31. Székely, G. J., Sen, P. K. (2002). Characterization of diagonal symmetry: location unknown, and a test based on allied U-processes. Journal of Statistical Planning and Inference, 102, 349–358.MathSciNetzbMATHGoogle Scholar
  32. Widder, D. V. (1959). The Laplace transform, 5th printing. Princeton: Princeton University Press.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2019

Authors and Affiliations

  1. 1.Institute of StochasticsKarlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations