Multivariate coefficients of variation: Comparison and influence functions

Abstract

In the univariate setting, coefficients of variation are well-known and used to compare the variability of populations characterized by variables expressed in different units or having really different means. When dealing with more than one variable, the use of such a relative dispersion measure is much less common even though several generalizations of the coefficient of variation to the multivariate setting have been introduced in the literature. In this paper, the lack of robustness of the sample versions of the multivariate coefficients of variation (MCV) is illustrated by means of influence functions and some robust counterparts based either on the Minimum Covariance Determinant (MCD) estimator or on the S estimator are advocated. Then, focusing on two of the considered MCV’s, a diagnostic tool is derived and its efficiency in detecting observations having an unduly large effect on variability is illustrated on a real-life data set. The influence functions are also used to compute asymptotic variances under elliptical distributions, yielding approximate confidence intervals. Finally, simulations are conducted in order to compare, in a finite sample setting, the performance of the classical and robust MCV’s in terms of variability and in terms of coverage probability of the corresponding asymptotic confidence intervals.