Data Plotting Methods for Checking Multivariate Normality and Related Ideas

Abstract

The object of the paper is to find directions along which multivariate observations have the greatest multivariate skewness or kurtosis in an appropriate sense. Typical measures of multivariate skewness and kurtosis are Malkovich and Afifi’s (1973) b1 * and b2 *, which are essentially nonlinear maximization problems. To avoid this nonlinearity we present an approach to reduce the above problems to easier ones, which are eigenvalue problems in linear algebra and closely related to some types of measures of multivariate skewness and kurtosis. By using the resultant directions we can project observations into the sample space and check normality of the data through probability plots and scatter plots. We also show that the proposed approach enables us to extend the usual principal component analysis to a higher order case.