A Q–Q plot for detecting non-multinormality based on a normal characterization and the S–W statistic

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.