1966/04/01 by Leon J. Gleser, Leon Jay Gleser · 2 citations
Mathematics · #Random Matrices and Applications #Statistical Methods and Bayesian Inference #Advanced Statistical Methods and Models
paper · pdf · doi:10.1214/aoms/1177699529
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 p(S) = Cp,n |S|(n - p - 1)/2 |Σ|-n/2 exp \lbrack -(1)/(2) tr Σ-1S\rbrack, S > 0 where C-1p,n = πp(p - 1)/42np/2 ∏pi = 1 Γ((n - i + 1)/2), p \leqq n, and n = N - 1. Henceforth we shall denote the fact that a random matrix Z has the density (1.1) by writing \mathfrakL(Z) = \mathfrakW(Σ, p, n); thus, \mathfrakL(S) = \mathfrakW(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: T(S) ≡ (tr S)p/|S| > K, where T(S)/pp is the -2/Nth 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.