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Tests of multinormality based on location vectors and scatter matrices

机译：基于位置向量和散射矩阵的多重正态性检验

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摘要

Classical univariate measures of asymmetry such as Pearson's (mean-median)/σ or (mean-mode)/σ often measure the standardized distance between two separate location parameters and have been widely used in assessing univariate normality. Similarly, measures of univariate kurtosis are often just ratios of two scale measures. The classical standardized fourth moment and the ratio of the mean deviation to the standard deviation serve as examples. In this paper we consider tests of multinormality which are based on the Mahalanobis distance between two multivariate location vector estimates or on the (matrix) distance between two scatter matrix estimates, respectively. Asymptotic theory is developed to provide approximate null distributions as well as to consider asymptotic efficiencies. Limiting Pitman efficiencies for contiguous sequences of contaminated normal distributions are calculated and the efficiencies are compared to those of the classical tests by Mardia. Simulations are used to compare finite sample efficiencies. The theory is also illustrated by an example.
机译：经典的不对称单变量测量方法，例如Pearson（均值）/σ或（均模）/σ，通常测量两个单独位置参数之间的标准距离，并已广泛用于评估单变量正态性。同样，单变量峰度的度量通常只是两个尺度度量的比率。经典的标准第四矩和平均偏差与标准偏差之比作为示例。在本文中，我们考虑基于两个多元位置向量估计之间的马氏距离或两个散布矩阵估计之间的（矩阵）距离的多重正态性检验。渐近理论的发展是为了提供近似零分布以及考虑渐近效率。计算了污染正态分布的连续序列的极限Pitman效率，并将其效率与Mardia的经典测试的效率进行了比较。仿真用于比较有限的样品效率。一个例子也说明了这一理论。

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