Statistical Methods and Applications

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

Tests of multinormality based on location vectors and scatter matrices

  • Annaliisa KankainenEmail author
  • Sara Taskinen
  • Hannu Oja
Original Article


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.


Affine invariance Kurtosis Pitman efficiency Skewness 


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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

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