Robustness and e ciency of multivariate coe cients of variation

The coe cient of variation is a well-known measure used in many elds to compare the variability of univariate variables having really di erent means or expressed in di erent scales. However, when the dimension of the problem is greater than one, comparing the variability only marginally may lead to controversial results. Several multivariate extensions of the univariate coe cient of variation have been introduced in the literature in order to summarize global relative variability in one single index (see Albert & Zhang, 2010, for a review). These multivariate coe cients are de ned in terms of the covariance matrix (via its trace, determinant,...) and of the mean vector of the underlying distribution. In practice, these coe cients can be estimated by plugging any pair of location and covariance estimators in their de nitions. However, as soon as the classical mean and covariance matrix are under consideration, the in uence functions are unbounded, while the use of any robust estimators yields bounded in uence functions. While useful in their own right, the in uence functions of the multivariate coe cients of variation will be further exploited in this talk to derive a general expression for the corresponding asymptotic variances under elliptical symmetry. Simulations compare the nite-sample e ciency of the classical estimator and the robust approach based on the Minimum Covariance Determinant estimator. Then, focusing on two of the considered multivariate coe cients, a diagnostic tool based on their in uence functions, as suggested in Pison & Van Aelst (2004) is derived. It allows to detect those observations having the tendency to increase or decrease the relative dispersion and it will be compared, on a real-life dataset, with the usual distance-plot.