A family of kurtosis orderings for multivariate distributions

In this paper, a family of kurtosis orderings for multivariate distributions is proposed and studied. Each ordering characterizes in an a¢ ne invariant sense the movement of probability mass from the "shoulders" of a distribution to either the center or the tails or both. All even moments of the Mahalanobis distance of a random vector from its mean (if exists) preserve a subfamily of the orderings. For elliptically symmetric distributions, each ordering determines the distributions up to a¢ ne equivalence. As applications, the orderings are used to study elliptically symmetric distributions. Ordering results are established for three important families of elliptically symmetric distributions: Kotz type distributions, Pearson Type VII distributions, and Pearson Type II distributions. AMS 2000 Subject Classi…cation: Primary 62G05 Secondary 62H05. Key words and phrases: Kurtosis; Peakedness; Tailweight; Ordering; Elliptically symmetric distributions. 1 Introduction and preliminaries Up to now, many multivariate kurtosis measures have been proposed (see, e.g., Mardia [10], Oja [13], Srivastava [16], Averous and Meste [1], Liu, Parelius and Singh [9], Ser‡ing [15], and Wang and Ser‡ing [18]). The classical notion of multivariate kurtosis is moment-based, given (Mardia [10]) by the fourth moment of the Mahalanobis distance of a random vector X in R from its mean , i.e., kd = E[(X ) 0 (X )]: kd measures the dispersion of X about the ellipsoid (x ) 0 (x ) = d, which de…nes the “shoulders”of the distribution. Higher kurtosis arises when probability mass is diminished near the shoulders and greater either near (greater peakedness), or greater in the tails (greater tailweight), or both. See Wang and Ser‡ing [18] for detailed discussion. Since the pioneering work of Bickel and Lehmann [3] [4] and Oja [12] about descriptive statistics, it has been commonly admitted that the meaning of a descriptive concept of distributions is given by an ordering and that measures for this concept are meaningful only if they preserve the ordering. While univariate kurtosis orderings and their applications have received considerable attention, kurtosis orderings for multivariate distributions have received relatively little investigation. There has been not even a multivariate kurtosis ordering for the classical multivariate 1 kurtosis measure kd up to now. Multivariate kurtosis measures are usually developed by intuition. It is necessary to study multivariate kurtosis by the ordering approach. That is the motivation of this work. For the univariate case, van Zwet [17] de…ned a kurtosis ordering s (s-ordering) for univariate symmetric distributions: FX s GY i¤ G 1 Y (FX(x)) is convex for x > F ; where F is the point of symmetry of FX . Using the folded distributions FjX X j and GjY Y j; Oja [12] gave an equivalent de…nition of the s-ordering: FX k GY i¤ G 1 jY Y j(FjX X j(z)) is convex for z 0; i.e., FjX X j c GjY Y j; where c is the van Zwet [17] skewness ordering for univariate distributions. This definition was extended by Balanda and MacGillivray [2] to include the case of univariate asymmetric distributions with …nite mean. They called FjX X j the moment–based spread function. To allow the use of other location measures instead of the mean X , we will call it the distribution-based spread function. Balanda and MacGillivray [2] also studied various univariate kurtosis orderings by the quantile–based spread function SF (p) = F 1( 2 + p 2 ) F 1( 2 p 2 ). In fact, the inverse function S 1 F of SF is a distribution function and can be considered as a distribution-based spread function. Extending S 1 F to the multivariate case, Averous and Meste [1] de…ned the multivariate kurtosis orderings in L1-sense.