Probability plots for assessing multivariate normality
Probability plots for assessing multivariate normality
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用于评估多元正态性的概率图
DOI：
10.2307/2348980
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
J. Koziol
中科院分区：
文献类型：
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作者：
J. Koziol
本文描述了一种通过将多变量偏度和峰度分解为正交分量来评估多变量正态性的方法。这些组件可以很容易地进行分析检查，以及与概率图图形。对单个成分的检查通常可以揭示偏离多元正态性的情况，这些情况可能在其潜在的偏度和峰度测量中被掩盖。存在许多评估多元正态性的技术;例如，参见Gnanadesikan（1977）、考克斯和Small（1978）、Mardia（1980）和Koziol（1986）的广泛综述。一种图形方法从“半径和角度”方法发展而来，该方法基于众所周知的欧几里德到球坐标的变换。Gnanadesikan（1977）特别描述了用图形程序检查球面坐标变换后的半径和角度的多元正态性的各种非正式检验; 03 The Dog（1978）（1982，1983）、Royston（1983）、Szkutnik（1987）以及Quiroz和达德利（1991）等人考虑了与这种方法相关的更正式的程序。用于评估多元正态性的不同范例是基于多元矩的属性。在一系列开创性的论文中（Mardia，1970，1974，1975; Mardia & Foster，1983; Mardia &金泽，1983），Mardia引入了多变量偏度和峰度的仿射不变测度，并全面考察了它们在各种情况下的性质。从另一个角度来看，Koziol（1986年，1987年）认为措施的多变量偏度和峰度建议的概念，内曼的顺利测试;在导出用于评估多元正态性的平滑检验后，他进一步发现多元偏斜度的平滑检验与Mardia的多元偏斜度测量一致，但多变量峰度的平滑检验与Mardia的多变量峰度测量略有不同。Koziol指出，平滑测试可以分解为单独的分量，这些分量作为独立的随机变量分布，每个变量都有一个自由度。另见Mardia（1987）以及Mardia和肯特（1991），他们从Rao的分数检验中得到了统一的处理方法。本说明的目的是描述成分的图形技术，并证明单个成分的检查可以是评估多元正态性的有价值的工具。为了完整起见，在第2节中简要回顾了平滑检验及其组成部分;使用组成部分评估多元正态性的示例见
A method of assessing multivariate normality by decomposition of measures of multivariate skewness and kurtosis into orthogonal components is described. The components can readily be inspected analytically, as well as graphically with probability plots. Examination of the individual components can often reveal departures from multivariate normality that may be masked in their underlying measures of skewness and kurtosis. There exist many techniques for assessing multivariate normality; see, for example, extensive reviews by Gnanadesikan (1977), Cox and Small (1978), Mardia (1980) and Koziol (1986). One graphical method devolves from the 'radius and angles' approach, which is based on the well-known transformation of Euclidean to spherical coordinates. Gnanadesikan (1977) in particular describes various informal tests for multivariate normal- ity with graphical procedures for examining the radii and angles following spherical coordinate transformation; Small (1978), Koziol (1982, 1983), Royston (1983), Szkutnik (1987) and Quiroz and Dudley (1991) among others have considered more formal procedures related to this approach. A different paradigm for assessing multivariate normality is based on the attributes of multivariate moments. In a series of seminal papers (Mardia, 1970, 1974, 1975; Mardia & Foster, 1983; Mardia & Kanazawa, 1983), Mardia introduced affine invariant measures of multivariate skewness and kurtosis, and comprehensively examined their properties under various circumstances. From a different perspective, Koziol (1986, 1987) considered measures of multivariate skewness and kurtosis suggested from the notion of Neyman's smooth tests; upon deriving smooth tests for assessing multivariate normality, he further found that the smooth test for multivariate skewness coincides with Mardia's measure of multivariate skewness, but the smooth test for multivariate kurtosis differed somewhat from Mardia's measure of multivariate kurtosis. Koziol noted that the smooth tests may be decomposed into individual components, which are distributed as independent %2 random variables, each with one degree of freedom. See also Mardia (1987) and Mardia and Kent (1991) for a unifying treatment devolving from Rao's score tests. The purpose of this note is to describe graphical techniques for depicting the components, and to demonstrate that examination of the individual components can be a valuable tool for assessing multivariate normality. For completeness, the smooth tests and their components are briefly reviewed in Section 2; examples of assessing multivariate normality with the components are found in