The joint distribution of Studentized residuals under elliptical distributions

Scaled and Studentized statistics are encountered frequently, and they often play a decisive role in statistical inference and testing. For instance, taking the sample mean vector X ̄ = ∑ j = 1 N X j / N and the sample covariance matrix S = ∑ j = 1 N ( X j − X ̄ ) ( X j − X ̄ ) ′ / ( N − 1 ) for an iid sample { X j } j = 1 N , some statistics for testing normality of the underlying distribution consist of the scaled residuals (the Studentized residuals or the transformed samples), Y j = S − 1 / 2 ( X j − X ̄ ) ( j = 1 , 2 , … , N ) . In this paper, the distribution of the random matrix the columns of which consist of the scaled residuals is derived under elliptical distributions. Also exact distributions of Studentized statistics are discussed as an application of the main result.