Testing the equality of several covariance matrices with fewer observations than the dimension

Abstract

Comparison of powers Equality of several covariance matrices Equality of two covariances High-dimensional data Normality Sample size smaller than the dimension a b s t r a c t For normally distributed data from the k populations with m × m covariance matrices Σ 1 , . . . , Σ k , we test the hypothesis H : Σ 1 = · · · = Σ k vs the alternative A = H when the number of observations N i , i = 1, . . . , k from each population are less than or equal to the dimension m, N i ≤ m, i = 1, . . . , k. Two tests are proposed and compared with two other tests proposed in the literature. These tests, however, do not require that N i ≤ m, and thus can be used in all situations, including when the likelihood ratio test is available. The asymptotic distributions of the test statistics are given, and the power compared by simulations with other test statistics proposed in the literature. The proposed tests perform well and better in several cases than the other two tests available in the literature.

A review of 20 years of naive tests of significance for high-dimensional mean vectors and covariance matrices

“…where Later, Srivastava and Yanagihara [45] and Srivastava et al [46] extended this work to the cases of two or more population covariance matrices and without normality assumptions, respectively. Furthermore, Cai and Ma [9] showed that T S1 is rate-optimal over this asymptotic regime, and Zhang et al [54] proposed the empirical likelihood ratio test for this problem.…”

A Bayesian framework for joint morphometry of surface and curve meshes in multi-object complexes

“…(Γ c α Γ p α ) Here, we assume that the initial momenta of patients and controls do not share the same covariance matrix. To test this assumption, we use the test proposed in Srivastava and Yanagihara (2010). The .…”

On testing the equality of mean vectors in high dimension

“…We also consider different scalar matrix In this section, we consider the problem of testing the equality of the mean vectors of two groups when the covariance matrices of the two groups are not equal. For normally distributed observation vectors, the equality of two covariance matrices can be ascertained using a test proposed by Srivastava and Yanagihara [13]. And if it is found that the covariance matrices of the two groups are not equal, the tests given in this section should be used.…”

“…Bai and Saranadasa's [1] test as well as the test proposed by Srivastava and Du [9] can be generalized to the case when the covariance matrices Σ 1 and Σ 2 of the two groups are not equal; for testing the equality of two covariance matrices, see Srivastava and Yanagihara [13]. The generalized versions of these two tests have been considered by Chen and Qin [2], and Srivastava, Katayama and Kano [11].…”