ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS
ESTIMATION OF MOMENT PARAMETER IN ELLIPTICAL DISTRIBUTIONS
复制标题
椭圆分布中矩参数的估计
DOI：
10.14490/jjss.33.215
复制
发表时间：
2003
期刊：
影响因子：
--
通讯作者：
T. Seo
中科院分区：
文献类型：
--
作者：
Yosihito Maruyama;T. Seo
作为一个典型的非正态情形，我们考虑一族椭圆对称分布。然后给出了矩参数及其相合估计。并给出了一般矩参数一致估计的渐近期望和渐近方差。此外，还提供了一些选定参数下的蒙特卡罗模拟结果。广义矩参数是椭圆总体多元统计分析研究中的重要峰度参数。峰度参数，特别是与估计问题有关的峰度参数，已经被许多作者所考虑。Mardia（1970，1974）定义了一个多变量样本峰度的度量，并导出了多变量正态总体样本峰度的渐近分布。同时，利用渐近结果考虑了检验的正态性。安德森（1993）和徐和富山（1996）给出了椭圆分布下峰度参数的相关讨论。Henze（1994）讨论了一般分布的穆尔样本峰度的渐近方差。本文讨论椭圆分布中一般矩参数的估计问题。特别地，我们推广了安德森（1993）的结果，推广了Mardia（1970，1974），Seo和富山（1996）的渐近性质.一般情况下，椭圆总体下检验统计量的精确分布或检验问题的最优解不容易求出，因此考虑了统计量的渐近展开。特别是高阶矩参数不仅包括峰度参数，而且还包括更一般的高阶矩参数。然后，我们必须推测的矩参数的估计作为一个实际问题。本文件的结构如下。首先介绍了椭圆分布的概率密度函数、特征函数和子类。其次，我们证明了有关矩的结果，并定义了矩
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