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(ν, p) |Σ|−1/2| [(ν − 2) + (x − θ)′ Σ−1(x − θ)]−(ν + p)/2, where θ and Σ denote the p × 1 location vector and p × p covariance matrix, respectively, and ν is the degrees of freedom of the distribution. This paper develops an asymptotically locally most powerful test for testing the covariance matrix Σ = Σ0, based on Neyman′s approach. The proposed test statistic has asymptotically χ2 distribution with γ degrees of freedom, where γ is the number of independent restrictions over the parameters, specified under the null hypothesis.