Some robust tests of independence in symmetrical multivariate distributions

Let X =(x)^ij=(11¹, …, X,)_T, i = l, …n, be an n X random matrix having multivariate symmetrical distributions with parameters μ, Σ. The p-variate normal with mean μ and covariance matrix is a member of this family. Let be the squared multiple correlation coefficient between the first and the succeeding p₁ components, and let p² = + be the squared multiple correlation coefficient between the first and the remaining p₁ + p₂ =p – 1 components of the p-variate normal vector. We shall consider here three testing problems for multivariate symmetrical distributions. They are (A) to test p² =0 against; (B) to test against =0, 0; (C) to test against p² =0, We have shown here that for problem (A) the uniformly most powerful invariant (UMPI) and locally minimax test for the multivariate normal is UMPI and is locally minimax as p² 0 for multivariate symmetrical distributions. For problem (B) the UMPI and locally minimax test is UMPI and locally minimax as for multivariate symmetrical distributions. For problem (C) the locally best invariant (LBI) and locally minimax test for the multivariate normal is also LBI and is locally minimax as for multivariate symmetrical distributions.