Moments and distributions of certain multivariate test criteria in the canonical correlation case under violation

Abstract

The joint probability density function (pdf) and the moments of order h of the general multivariate statistic defined by Y (p) (a,b) = П p i = 1 ℓ a i(1 - ℓ i) b , where a and b are positive real numbers, are obtained in this paper in the canonical correlation case and expressed in terms of the H-function. Alternative forms of some test criteria are also furnished in terms of Meijer G-function. Explicit forms of the pdf and the cdf are also supplemented in the case p = 2. Here ℓ 1, ℓ 2, …, ℓ p are the latent roots of the matrix L = λR(1 + λR) −1, R = S 1 S −1 2, and λ > 0. The covariance matrices S 1 and S 2 are independently distributed and have, respectively, the noncentral Wishart distribution W( p, n 1, Σ 1, Щ) , Щ = 1 2 MM′Σ −1 1 and the central Wishart distribution W( p, n 2, Σ 2, 0). The derivation of this pdf is based on Pillai's density function of S 1 S −1 2 under violation. It is necessary to point out that violation means the assumption of normality is violated and the common covariance matrix is disturbed. Furthermore, the moments of order h and the pdf of some test criteria such as, Wilks, Wilks-Lawely and the modified likelihood ratio may be deduced from the results associated with the general statistic as special cases. The results of this paper may be used further to investigate the exact robustness against nonnormality of the test of independence.