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

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 < tr Σ i / p < ∞ as p → ∞ for i = 1 , … , 16 and (B) p / n → c < ∞ known as 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.