Score Test for the Covariance Matrix of the Elliptic t-Distribution

Let x1, ..., xj, ..., xn be n independent realizations of a p-dimensional random variable X which has the elliptical t-distribution of the form g(x) = K([nu], p) [Sigma]-1/2 [([nu] - 2) + (x - [theta])' [Sigma]-1(x - [theta])]-([nu] + p)/2, where [theta] and [Sigma] denote the p - 1 location vector and p - p covariance matrix, respectively, and [nu] is the degrees of freedom of the distribution. This paper develops an asymptotically locally most powerful test for testing the covariance matrix [Sigma] = [Sigma]0, based on Neyman's approach. The proposed test statistic has asymptotically [chi]2 distribution with [gamma] degrees of freedom, where [gamma] is the number of independent restrictions over the parameters, specified under the null hypothesis.