A Note on the Sphericity Test

Let x be a random p × 1 column vector having a multivariate normal distribution with unknown mean vector μ and unknown covariance matrix Σ. We wish to test the hypothesis of "sphericity," namely H:Σ = σ²Iₚ, where σ² > 0 is an unknown positive constant. Alternatives to H which are considered are HA : Σ positive definite, but Σ ≠ σ²I. Given N observation vectors x⁽¹⁾, x⁽²⁾, ⋯, x(N), independently distributed, each with the distribution of x, we can reduce consideration to the sufficient statistic ( x, S), where x = N⁻¹ ∑Ni = 1 x⁽ⁱ⁾, S = ∑Ni = 1 (x⁽ⁱ⁾ - x)(x⁽ⁱ⁾ - x)'. Then x has a multivariate normal distribution with mean vector μ and covariance matrix Σ/N, and S has the Wishart distribution, i.e., has density {equation*}{1.1}p(S) = Cp,n |S|(n - p - 1)/2 |Σ|-n/2 exp -1/2 tr Σ⁻¹S, S > 0{equation*} where C⁻¹p,n = πp(p - 1)/42np/2 ∏ᵖi = 1 Γ((n - i + 1)/2), p n, and $n = N - 1$. Henceforth we shall denote the fact that a random matrix Z has the density (1.1) by writing L(Z) = W(Σ, p, n); thus, L(S) = W(Σ, p, n). Mauchly [4] has found the likelihood ratio test for H v.s. HA. The rejection region of this test can be written in the form: {equation*}{1.2}T(S) ≡ (tr S)^p/|S| > K,{equation*} where T(S)/pᵖ is the $-2/N$th power of the likelihood ratio statistic λ. The moments of the likelihood ratio statistic λ under H were obtained by Mauchly [4]. Anderson [1] uses these moments to give the exact distribution of λ under H and to obtain an asymptotic expansion of this null distribution. The distribution of λ under HA has been obtained for the case $p = 2$ by Girshick [3], but the distribution of λ under HA for $p > 2$ appears to be highly untractable. In this note, we show that the distribution of $T(S)$ is related to the distribution of Bartlett's statistic for testing homogeneity of variances (viz., Anderson [1]). From this relation, we derive that Mauchly's test (1.2) is unbiased. A derivation of the asymptotic distribution of $T(S)$ under HA completes the note. It should be mentioned here that a direct relationship between the likelihood ratio statistic λ and the Bartlett statistic for testing the homogeneity of variances for the elements of x is given by Anderson [1]. He shows that H:Σ = σ²I is a combination of two hypotheses H₁:Σ is diagonal, and H₂:Σ = σ²I given that Σ is diagonal. Hypothesis H₂ is the hypothesis of the homogeneity of the variances of the elements of the vector x given that these random elements are stochastically independent. The likelihood ratio statistic λ₂ for testing this hypothesis is a monotone function of Bartlett's statistic. Further, the likelihood ratio statistic λ for H is the product λ = λ₁λ₂ of λ₂ and the likelihood ratio statistic λ₁ for testing H₁ (Anderson [1], pp. 260-2). Unfortunately, both the distribution of λ₁ and the distribution of λ₂ depend upon the unknown Σ, and, unless Σ is diagonal, λ₁ and λ₂ are dependent. As a result, this relationship between λ and Bartlett's statistic is difficult to exploit in finding the distribution of λ. In this note, we use the invariance of λ under orthogonal transformations to enable us to change to new variables having a diagonal covariance matrix. Homogeneity of variances for these new variables is shown to be equivalent to H:Σ = σ²I under the old variables. Using Anderson's representation for H, but now expressed in terms of the new variables, we have λ = λ₁'λ₂', where λ₂' tests homogeneity of variances for the new variables and λ₁' tests the diagonality of the new covariance matrix. Since the new covariance matrix is diagonal, λ₁' and λ₂' are independent and λ₁' has a distribution independent of the parameters. Such a representation is (hopefully) convenient for determining the properties of the likelihood ratio test based on λ. This representation, however, only connects the distribution of λ and the distribution of Bartlett's statistic, for the "new" variables used in our representation are not observable, but rather are functions of the unknown covariance matrix Σ.