An overall test for multivariate normality

Abstract

There are a number of methods in the statistical literature for testing whether observed data came from a multivariate normal(MVN) distribution with an unknown mean vector and covariance matrix. Let X₁, ...,X_n be an iid sample of size n from a p-variate normal distribution. Denote the sample mean and sample variance-covariance matrix by X̅ and S respectively. Most of the tests of multivariate normality are based on the results that Y_i=S^-½(X_i-X̅), i=1,..., n, are asymptotically iid as p-variate normal than zero mean vector and identity covariance matrix. Tests developed by Andrews et al., Mardina and others are direct functions of Y_i. We note that the N=np components of the Y_i's put together can be considered as an asymptotically iid sample of size N from a univariate normal any well known test based on N independent observations for univariate normality. In Particular we can use univariate skewness and kurtosis tests, which are sensitive to deviations from normality.