Abstract

A quantile-quantile (Q–Q) plot is derived from a characterization for the multivariate normal distribution and the Shapiro–Wilk’s (1965) S–W statistic. The normal characterization results in some independent spherical distributions. The affine invariance of the S–W statistic and a simple property of spherical distributions are employed to construct the Q–Q plot. Easy simulation of the empirical distribution of the S–W statistic avoids the complicated exact null distribution of S-W statistic. The Q–Q plot can be easily implemented for detecting a possible departure from multivariate normality in high dimensional data analysis. Two examples are illustrated for real application.

摘要

分位数-分位数 (Q-Q) 图源自多元正态分布的表征和 Shapiro-Wilk (1965) S-W 统计量。正态表征导致一些独立的球形分布。使用 S-W 统计量的仿射不变性和球形分布的简单属性来构建 Q-Q 图。轻松模拟 S-W 统计量的经验分布，避免了 SW 统计量复杂的精确零分布。Q-Q 图可以很容易地实现，以检测高维数据分析中可能偏离多变量正态性。为实际应用举例说明了两个例子。