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A New Statistic for Testing Multivariate Normality

One generally assumes multivariate normality in most of the multivariate statistical procedures and techniques related with the assessment of this underlying assumption has always received considerable attention. Unfortunately, in practice, testing for multivariate normality is more diffucult than testing for univariate normality and relatively few formal methods are available in this context. Mardia (1970) proposes a test of multivariate normality based on some measures of multivariate skewness and kurtosis. Cox and Small (1978) reviews some of the earlier work on testing multivariate normality and develops both coordinate-dependent and invariant tests. Various formal and informal procedures proposed for testing the hypothesis of multivariate normality were extensively reviewed by Gnanadesikan (1977,Section 5.4.2). Small (1980) combined the sample marginal skewness and kurtosis via Johnson Su transformations to produce a statistic for testing multivariate normality. Ozturk and Romeu (1992) transforms the multivariate sample into a set of uncorrelated samples and represents the order statistics of each transformed sample by linked vectors in a two dimensional space to develop a procedure for testing multivariate normality. In general, tests for multinormality can be grouped into four categories ,i.e., empirical distribution function tests, correlation tests, empirical characteristic function tests and procedures based on measures of skewness and kurtosis. Assume now that the p-dimensional random vectors X1,...,Xn are n independent observations from a multivariate normal distribution with mean vector μ and covariance matrix Σ. The squared Mahalonobis distance between Xi and μ is then given by