Testing homogeneity of several covariance matrices and multi-sample sphericity for high-dimensional data under non-normality

ABSTRACT

A test for homogeneity of g ⩾ 2 covariance matrices is presented when the dimension, p, may exceed the sample size, n_i, i = 1, …, g, and the populations may not be normal. Under some mild assumptions on covariance matrices, the asymptotic distribution of the test is shown to be normal when n_i, p → ∞. Under the null hypothesis, the test is extended for common covariance matrix to be of a specified structure, including sphericity. Theory of U-statistics is employed in constructing the tests and deriving their limits. Simulations are used to show the accuracy of tests.

KEYWORDS:

• High-dimensionality
• multi-sample sphericity
• non-normality
• U-statistics.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H15

Acknowledgments

Sincere thanks to the editor, the associate editor and two anonymous referees whose comments helped significantly improve the paper.