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Tests of multinormality based on location vectors and scatter matrices

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Jyväskylä, 40014, Jyväskylä, Finland

    Annaliisa Kankainen & Sara Taskinen

  2. Tampere School of Public Health, University of Tampere, 33014, Tampere, Finland

    Hannu Oja

Authors
  1. Annaliisa Kankainen
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  2. Sara Taskinen
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  3. Hannu Oja
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Correspondence to Annaliisa Kankainen.

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Kankainen, A., Taskinen, S. & Oja, H. Tests of multinormality based on location vectors and scatter matrices. Stat. Meth. & Appl. 16, 357–379 (2007). https://doi.org/10.1007/s10260-007-0045-9

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  • DOI: https://doi.org/10.1007/s10260-007-0045-9

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