Robustness and efficiency of multivariate coefficients of variation

The coefficient of variation is a well-known measure used in many fields to compare the variability of a variable in several populations. However, when the dimension is greater than one, comparing the variability only marginally may lead to controversial results. Several multivariate extensions of the univariate coefficient of variation have been introduced in the literature. In practice, these coefficients can be estimated by using any pair of location and covariance estimators. However, as soon as the classical mean and covariance matrix are under consideration, the influence functions are unbounded, while the use of any robust estimators yields bounded influence functions.
While useful in their own right, the influence functions of the multivariate coefficients of variation are further exploited in this talk to derive a general expression for the corresponding asymptotic variances under elliptical symmetry. Then, focusing on two of the considered multivariate coefficients, a diagnostic tool based on their influence functions is derived and compared, on a real-life dataset, with the usual distance-plot.