Testing a block exchangeable covariance matrix

ABSTRACT

Testing hypotheses about the structure of a covariance matrix for doubly multivariate data is often considered in the literature. In this paper the Rao's score test (RST) is derived to test the block exchangeable covariance matrix or block compound symmetry (BCS) covariance structure under the assumption of multivariate normality. It is shown that the empirical distribution of the RST statistic under the null hypothesis is independent of the true values of the mean and the matrix components of a BCS structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Simulation studies are performed for the sample size consideration, and for the estimation of the empirical quantiles of the null distribution of the test statistic. The RST procedure is illustrated on a real data set from the medical studies.

KEYWORDS:

• Block compound symmetry covariance structure
• block exchangeable covariance matrix
• hypothesis testing
• maximum likelihood estimates
• Rao's score test

AMS SUBJECT CLASSIFICATION:

• 62H15
• 62H12

Acknowledgments

The authors thank to the editor and two anonymous reviewers for all the valuable comments and suggestions. The authors also thank Mathematical Research and Conference Center of the Polish Academy of Sciences for giving the opportunity and support for this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is partially supported by Statutory Activities (Politechnika Poznańska) [No. 04/43/DSPB/0085] (Katarzyna Filipiak) and by grant Ministerstvo školstva, vedy, výskumu a športu Slovenskej republiky [VEGA MS SR 1/0073/15] (Daniel Klein).