Doornik-Hansen (1994) introduce a multivariate version of the univariate omnibus test for normality of Shenton and Bowman (1977), based on the transformed skewness and kurtosis. Due that the skewness and kurtosis are not independently distributed and the kurtosis approaches very slowly to normality, they propose a test assuming independence of skewness and kurtosois. The kurtosis is less convenient for it is conditional to more than 1 + skewness, and with a gamma distribution. So, they suggest to transform them to create statistics much closer to standard normal. Where the transformation for the skewness is as in D'Agostino (1970), and the kurtosis is tranformed from a gamma to a Chi-square distribution using the Wilson-Hilferty cubed root transformation, which then is translated into standard normal.
Using the correlation matrix (makes the test scale invariant) and the diagonal matrix of reciprocals of the p standard deviations, a multivariate normal can thus be transformed into independent standard normals.
The skewness is transformed using the D'Agostino procedure, and the kurtosis is transformed from a gamma distribution to a chi-square. The the Doornik-Hansen (DH) statistic can approximates to a chi-square with 2p degrees of freedom.
If the rank of the correlation matrix is less than p, some eigenvalues will be zero. In that case the following procedure can be used. Select the eigenvectors corresponding to the p* non-nonzero eigenvalues, G say, and create a new data matrix X* = GX. This will be an n x p* matrix. Now compute DH using X* and base the tests on p* degrees of freedom.
This Doornik-Hansen omnibus test can test a multivariate or univariate normal distribution.
X - data matrix (Size of matrix must be n-by-p; data=rows, indepent variable=columns)
alpha - significance level (default = 0.05)
- Doornik-Hansen Omnibus Multivariate (Univariate) Normality Test