Abstract
As a typical non-normal case, we consider a family of elliptically symmetric dis- tributions. Then, the moment parameter and its consistent estimator are presented. Also, the asymptotic expectation and the asymptotic variance of the consistent es- timator of the general moment parameter are given. Besides, the numerical results obtained by Monte Carlo simulation for some selected parameters are provided. The general moment parameter includes the important kurtosis parameter in the study of multivariate statistical analysis for elliptical populations. The kurtosis parameter, especially with relation to the estimation problem, has been considered by many authors. Mardia (1970, 1974) defined a measure of multi- variate sample kurtosis and derived its asymptotic distribution for samples from a multivariate normal population. Also, the testing normality was considered by using the asymptotic result. The related discussion of the kurtosis parameter under the elliptical distribution has been given by Anderson (1993), and Seo and Toyama (1996). Henze (1994) has discussed the asymptotic variance of the mul- tivariate sample kurtosis for general distributions. Here we deal with the estima- tion of the general moment parameters in elliptical distributions. In particular, we make a generalization of the results of Anderson (1993) and give an extension of asymptotic properties in Mardia (1970, 1974), Seo and Toyama (1996). In general, it is not easy to derive the exact distribution of test statistics or the percentiles for the testing problem under the elliptical populations, and so the asymptotic expansion of the statistics is considered. Especially that given up to the higher order includes not only the kurtosis parameter but the more gen- eral higher order moment parameters as well. Then we have to speculate about the estimation of the moment parameters as a practical problem. The present paper is organized in the following way. First, the probability density function, characteristic function and subclasses of the elliptical distribution are explained. Secondly, we prove the results concerning the moments and define the moment
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