Distribution and robustness of a distance-based multivariate coefficient of variation

When one wants to compare the homogeneity of a characteristic in several popula-
tions that have di erent means, the advocated statistic is the univariate coe cient of variation.
However, in the multivariate setting, comparing marginal coe cients may be inconclusive.
Therefore, several extensions that summarize multivariate relative dispersion in one single in-
dex have been proposed in the literature (see Albert & Zhang, 2010, for a review).
In this poster, focus is on a particular extension, due to Voinov & Nikulin (1996), based on the
Mahalanobis distance between the mean and the origin of the design space. Some arguments
are outlined for justifying this choice. Then, properties of its sample version under elliptical
symmetry are discussed. Under normality, this estimator is shown to be biased at nite samples.
In order to overcome this drawback, two bias corrections are proposed.
Moreover, the empirical estimator also su ers from a lack of robustness, which is illustrated
by means of in uence functions. A robust counterpart based on the Minimum Covariance
Determinant estimator is advocated.