Asymptotics for tests on mean profiles, additional information and dimensionality under non-normality

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5 Scopus citations


We consider the comparison of mean vectors for k groups when k is large and sample size per group is fixed. The asymptotic null and non-null distributions of the normal theory likelihood ratio, Lawley-Hotelling and Bartlett-Nanda-Pillai statistics are derived under general conditions. We extend the results to tests on the profiles of the mean vectors, tests for additional information (provided by a sub-vector of the responses over and beyond the remaining sub-vector of responses in separating the groups) and tests on the dimension of the hyperplane formed by the mean vectors. Our techniques are based on perturbation expansions and limit theorems applied to independent but non-identically distributed sequences of quadratic forms in random matrices. In all these four MANOVA problems, the asymptotic null and non-null distributions are normal. Both the null and non-null distributions are asymptotically invariant to non-normality when the group sample sizes are equal. In the unbalanced case, a slight modification of the test statistics will lead to asymptotically robust tests. Based on the robustness results, some approaches for finite approximation are introduced. The numerical results provide strong support for the asymptotic results and finiteness approximations.

Original languageEnglish
Pages (from-to)2685-2705
Number of pages21
JournalJournal of Statistical Planning and Inference
Issue number8
StatePublished - Aug 1 2009

Bibliographical note

Funding Information:
This research was supported by the Montana Board of Research and Commercialization Technology (MBRCT) Grant no. NSF EPSCoR 06-07. The author is grateful to the editors and reviewers for their helpful comments. The author would also like to thank Professors Arne C. Bathke and Brian Steele for their valuable suggestions on the early draft of the manuscript. Appendix A


  • Asymptotics
  • Dimensionality
  • Distribution of eigenvalues
  • Perturbation expansion
  • Robustness
  • Tests for additional information

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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