Distribution under elliptical symmetry of a distance-based multivariate coefficient of variation

In the univariate setting, the coefficient of variation is widely used to measure the relative dispersion of a random variable with respect to its mean. Several extensions of the univariate coefficient of variation to the multivariate setting have been introduced in the literature. In this paper, we focus on a distance-based multivariate coefficient of variation. First, some real examples are discussed to motivate the use of the considered multivariate dispersion measure. Then, the asymptotic distribution of several estimators is analyzed under elliptical symmetry and used to construct approximate parametric confidence intervals that are compared with non-parametric intervals in a simulation study. Under normality, the exact distribution of the classical estimator is derived. As this natural estimator is biased, some bias corrections are proposed and compared by means of simulations.