Spearman rank correlation of the bivariate Student t and scale mixtures of normal distributions

Abstract

We derive an expression for the Spearman rank correlation of bivariate scale mixtures of normals (SMN) and we show that within this class, for any value of the correlation parameter, the Spearman rank correlation of the normal is the greatest in absolute value. We then provide expressions for the symmetric generalized hyperbolic, the Bessel, and the Laplace distributions. We further derive an expression for the Spearman rank correlation of the Student t distribution in terms of an easily computable one-dimensional integral, and we also consider the special case of the Cauchy. Finally, we show how our results can be used in a rank-based estimation of the parameters of the Student t distribution.