Advertisement

Statistical Methods and Applications

, Volume 16, Issue 3, pp 357–379 | Cite as

Tests of multinormality based on location vectors and scatter matrices

  • Annaliisa Kankainen
  • Sara Taskinen
  • Hannu Oja
Original Article

Abstract

Classical univariate measures of asymmetry such as Pearson’s (mean-median)/σ or (mean-mode)/σ often measure the standardized distance between two separate location parameters and have been widely used in assessing univariate normality. Similarly, measures of univariate kurtosis are often just ratios of two scale measures. The classical standardized fourth moment and the ratio of the mean deviation to the standard deviation serve as examples. In this paper we consider tests of multinormality which are based on the Mahalanobis distance between two multivariate location vector estimates or on the (matrix) distance between two scatter matrix estimates, respectively. Asymptotic theory is developed to provide approximate null distributions as well as to consider asymptotic efficiencies. Limiting Pitman efficiencies for contiguous sequences of contaminated normal distributions are calculated and the efficiencies are compared to those of the classical tests by Mardia. Simulations are used to compare finite sample efficiencies. The theory is also illustrated by an example.

Keywords

Affine invariance Kurtosis Pitman efficiency Skewness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bera A, John S (1983) Tests for multivariate normality with Pearson alternatives. Comm Statist Theory Methods 12:103–117CrossRefGoogle Scholar
  2. Cox DR, Small NJH (1978) Testing multivariate normality. Biometrika 65:263–272MATHCrossRefGoogle Scholar
  3. Davies PL (1987) Asymptotic behavior of S-estimates of multivariate location parameters and dispersion matrices. Ann Statist 15:1269–1292MATHMathSciNetGoogle Scholar
  4. Field C (1993) Tail areas of linear combinations of chi-squares and non-central chi-squares. J Stat Comp Simul 45:243–248MATHCrossRefGoogle Scholar
  5. Geary RC (1935) The ratio of the mean deviation to the standard deviation as a test of normality. Biometrika 27(3/4):310–332MATHCrossRefGoogle Scholar
  6. Gnanadesikan R (1977) Methods for statistical data analysis of multivariate observations. Wiley, New YorkMATHGoogle Scholar
  7. Henze N (1997) Limit laws for multivariate skewness in the sense of Móri, Rohatgi and Székely. Stat Probab Lett 33:299–307MATHCrossRefMathSciNetGoogle Scholar
  8. Hettmansperger TP, Randles RH (2002) A practical affine equivariant multivariate median. Biometrika 89:851–860MATHCrossRefMathSciNetGoogle Scholar
  9. Huber PJ (1981) Robust statistics. Wiley, New YorkMATHGoogle Scholar
  10. Kankainen A, Taskinen S, Oja H (2004) On Mardia’s tests of multinormality. In: Hubert M, Pison G, Stryuf A, Van Aelst S (eds) Statistics for industry and technology. Birkhauser, Basel, pp 153–164Google Scholar
  11. Koziol JA (1982) A class of invariant procedures for assessing multivariate normality. Biometrika 69:423–427MATHCrossRefMathSciNetGoogle Scholar
  12. Koziol JA (1983) On assessing multivariate normality. J R Statist Soc Ser B 45:358–361MATHMathSciNetGoogle Scholar
  13. Koziol JA (1986) Assessing multivariate normality: a compendium. Comm Statist Theory Methods 15:2763–2783MATHCrossRefMathSciNetGoogle Scholar
  14. Koziol JA (1987) An alternative formulation of Neyman’s smooth goodness of fit tests under composite alternatives. Metrika 34:17–24MATHCrossRefMathSciNetGoogle Scholar
  15. Koziol JA (1993) Probability plots for assessing multivariate normality. The Statistician 42:161–173CrossRefGoogle Scholar
  16. Lopuhaä HP (1989) On the relation between S-estimators and M-estimators of multivariate location and covariance. Ann Statist 17:1662–1683MATHMathSciNetGoogle Scholar
  17. MacGillvray HL (1986) Skewness and asymmetry: measures and orderings. Ann Statist 14:994–1011MathSciNetGoogle Scholar
  18. Malkovitch JF, Afifi AA (1973) On tests for multivariate normality. J Am Statist Assoc 68:176–179CrossRefGoogle Scholar
  19. Mardia KV (1970) Measures of multivariate skewness and kurtosis with applications. Biometrika 57:519–530MATHCrossRefMathSciNetGoogle Scholar
  20. Mardia KV (1974) Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies. Sankhyã Ser B 36(2):115–128MATHMathSciNetGoogle Scholar
  21. Mardia KV (1980) Tests of univariate and multivariate normality. Handbook Statist 1:279–320Google Scholar
  22. Maronna RA (1976) Robust M-estimators of multivariate location and scatter. Ann Statist 4:51–67MATHMathSciNetGoogle Scholar
  23. Móri TF, Rohatgi VK, Székely GJ (1993) On multivariate skewness and kurtosis. Theor Probab Appl 38:547–551CrossRefGoogle Scholar
  24. Nyblom J, Mäkeläinen T (1983) Comparisons of tests for the presence of random walk coefficients in a simple linear model. J Am Statist Assoc 78:856–864MATHCrossRefGoogle Scholar
  25. Oja H (1981) On location, scale, skewness and kurtosis of univariate distributions. Scand J Statist 8:154–168MathSciNetGoogle Scholar
  26. Ollila E, Hettmansperger TP, Oja H (2003) Affine equivariant multivariate sign methods. Under revisionGoogle Scholar
  27. Pearson K (1895) Contributions to the mathematical theory of evolution II. Skew variation in homogeneous material. Phil. Trans. R. Soc. Lond. A 186:343–414CrossRefGoogle Scholar
  28. Rao CR (1965) Linear statistical inference and its applications. Wiley, New YorkMATHGoogle Scholar
  29. Romeu JL, Ozturk A (1993) A comparative study of goodness-of-fit tests for multivariate normality. J Multivar Anal 46:309–334MATHCrossRefMathSciNetGoogle Scholar
  30. Rousseeuw PJ, Leroy AM (1987) Robust regression and outlier detection. Wiley, New YorkMATHGoogle Scholar
  31. Tyler DE (1982) Radial estimates and the test for sphericity. Biometrika 69:429–436MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland
  2. 2.Tampere School of Public HealthUniversity of TampereTampereFinland

Personalised recommendations