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Testing the hypothesis of a doubly exchangeable covariance matrix


In this paper the authors study the problem of testing the hypothesis of a doubly exchangeable covariance matrix for three-level multivariate observations, taken on m variables over u sites and over v time/space points. Through the decomposition of the main hypothesis into a set of three sub-hypotheses, the likelihood ratio test statistic is defined, its exact moments are determined, and its exact distribution is studied. Because this distribution is very much intricate, a very precise near-exact distribution is developed. Numerical studies conducted to evaluate the closeness between this near-exact distribution and the exact distribution show the very good performance of this approximation even for very small sample sizes. A simulation study is also conducted and two real-data examples are presented.

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This research was partially supported by CMA/FCT/UNL, under projects UID/MAT/00297/2013 and UID/MAT/00297/2019. The second author thanks the support for the summer research Grant from the College of Business at the University of Texas at San Antonio. The authors also want to thank the Editor-in-Chief and the two anonymous reviewers for their careful reading, valuable comments and suggestions that led to a quite improved version of the manuscript.

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Correspondence to Carlos A. Coelho.

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This research was partially supported by FCT–Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology), projects UID/MAT/00297/2013 and UID/MAT/00297/2019, through Centro de Matemática e Aplicações (CMA/FCT/UNL). The second author also thanks the support from the summer research grant from the College of Business at the University of Texas at San Antonio.

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Coelho, C.A., Roy, A. Testing the hypothesis of a doubly exchangeable covariance matrix. Metrika 83, 45–68 (2020).

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  • Characteristic function
  • Composition of hypotheses
  • Distribution of likelihood ratio statistics
  • Mixtures
  • Near-exact distributions
  • Product distribution

Mathematics Subject Classification

  • 62H15
  • 62H10
  • 62E15
  • 62E20
  • 62E10
  • 60E10