A necessary test of fit of specific elliptical distributions based on an estimator of Song’s measure

In a recent paper, Zografos [K. Zografos, On Mardia’s and Song’s measures of kurtosis in elliptical distributions, J. Multivariate Anal. 99 (2008) 858–879] has obtained general formulas for Song’s measure for the elliptic family of distributions, and he introduced and studied its sample analogue. In this paper, based on the empirical estimator of this measure, we present a test to verify if the data are distributed according to a specific elliptical (spherical) distribution. In this context, the asymptotic distribution of the proposed statistic under the null hypothesis of specific spherical distributions is obtained. The proposed statistic also provides us with a procedure for testing multivariate normality. In order to evaluate the convergence of the proposed statistic to its limiting distribution, under the null hypothesis, a simulation study is performed to analyze the behavior of the percentiles of the proposed statistic in some special cases of spherical distributions. Moreover, a Monte Carlo study is carried out on the performance of the test statistic as a necessary test of fit of specific spherical distributions. In this framework, the type I error rates as well as the power of the test are studied. Finally, a well-known data set is used to illustrate the method developed in this paper.