A new test for sphericity of the covariance matrix for high dimensional data

AMS subject classifications: 62H10 62H15 Keywords: Covariance matrix Hypothesis testing High-dimensional data analysis a b s t r a c t In this paper we propose a new test procedure for sphericity of the covariance matrix when the dimensionality, p, exceeds that of the sample size, N = n + 1. Under the assumptions that (A) 0 < trΣ the concentration, a new statistic is developed utilizing the ratio of the fourth and second arithmetic means of the eigenvalues of the sample covariance matrix. The newly defined test has many desirable general asymptotic properties, such as normality and consistency when (n, p) → ∞. Our simulation results show that the new test is comparable to, and in some cases more powerful than, the tests for sphericity in the current literature. Many applications of modern multivariate statistics involve a large number of variables, p, and hence a large covariance matrix. In many situations (e.g. DNA microarray data) the dimensionality exceeds that of the number of observations, N = n +1. In this article, we discuss much of the previous work in developing statistics for testing if the covariance matrix is proportional to the identity, more commonly called Sphericity. We consider X 1 ,. .. , X N as a set of independent observations from a multivariate normal distribution N p (µ, Σ), where both the mean vector µ ∈ R p and covariance matrix Σ > 0 are unknown. We are interested in testing H 0 : Σ = σ 2 I vs. H A : Σ = σ 2 I, where σ 2 is the unknown scalar proportion. The classical hypothesis testing techniques are based on the likelihood ratio and are degenerate when p > n. Motivated by the previous work in the literature, we define a new test statistic under the framework known as general asymptotics or (n, p)-asymptotics. Much of the current work rests on the large body of literature regarding asymptotics for eigenvalues of the sample Yin and Krishnaiah [24] and others. We build on the substantial list of work completed on statistical testing involving large random matrices, such as Bai et al. [4], Saranadasa [13] and most recently the work completed by Ledoit and Wolf [11], Srivastava [21–23] and Schott [14–16]. Ledoit and Wolf [11] show the locally best invariant test based on John's U statistic, see [10], to be (n, p)-consistent when p/n → c …