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On the limiting distribution of sample central moments
Annals of the Institute of Statistical Mathematics ( IF 1 ) Pub Date : 2018-11-28 , DOI: 10.1007/s10463-018-0695-4
Georgios Afendras , Nickos Papadatos , Violetta E. Piperigou

We investigate the limiting behavior of sample central moments, examining the special cases where the limiting (as the sample size tends to infinity) distribution is degenerate. Parent (non-degenerate) distributions with this property are called singular , and we show in this article that the singular distributions contain at most three supporting points. Moreover, using the delta -method, we show that the (second-order) limiting distribution of sample central moments from a singular distribution is either a multiple, or a difference of two multiples of independent Chi-square random variables with one degree of freedom. Finally, we present a new characterization of normality through the asymptotic independence of the sample mean and all sample central moments.

中文翻译:

关于样本中心矩的极限分布

我们研究样本中心矩的极限行为,检查极限(因为样本量趋于无穷大)分布退化的特殊情况。具有此属性的父(非退化)分布称为奇异分布,我们在本文中展示奇异分布最多包含三个支持点。此外,使用 delta 方法,我们表明来自奇异分布的样本中心矩的(二阶)极限分布是具有一个自由度的独立卡方随机变量的两个倍数的倍数或差值. 最后,我们通过样本均值和所有样本中心矩的渐近独立性来呈现正态性的新特征。
更新日期:2018-11-28
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