Multivariate statistical simulation
Multivariate statistical simulation
Elliptically contoured models in statistics
Elliptically contoured models in statistics
A necessary test of goodness of fit for sphericity
Journal of Multivariate Analysis
A test for elliptical symmetry
Journal of Multivariate Analysis
On Mardia's and Song's measures of kurtosis in elliptical distributions
Journal of Multivariate Analysis
Testing for ellipsoidal symmetry: A comparison study
Computational Statistics & Data Analysis
Multivariate measures of skewness for the skew-normal distribution
Journal of Multivariate Analysis
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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.