Doornik-Hansen test

The Doornik-Hansen test for multivariate normality (DOORNIK, J.A., and HANSEN, H. (2008)) is based on the skewness and kurtosis of multivariate data that is transformed to ensure independence. It is more powerful than the Shapiro-Wilk test for most tested multivariate distributions1.

The skewness and kurtosis are defined, respectively, as: and $k = \frac{m_{4}}{m_{2}^{2}},\ $where: $m_{i} = \frac{1}{n}\sum_{i = 1}^{n}{(x_{i}}{- \overline{x})}^{i}$ $\overline{x} = \frac{1}{n}\sum_{i = 1}^{n}x_{i}$ and $n$ is a number of (non-missing) residuals.

The Doornik-Hansen test statistic derives from SHENTON, L.R., and BOWMAN, K.O. (1977) and uses transformed versions of skewness and kurtosis.

The transformation for the skewness $s$ into$\text{z}_{1}$ is as in D'AGOSTINO, R.B. (1970):

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The kurtosis $k$ is transformed from a gamma distribution to $\chi^{2}$, which is then transformed into standard normal $z_{2}$ using the Wilson-Hilferty cubed root transformation:

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Finally, the Doornik-Hansen test statistic is defined as the sum of squared transformations of the skewness and kurtosis. Approximately, the test statistic follows a $\chi^{2}$distribution, i.e.:

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  1. The description of the test derives from DOORNIK, J.A., and HANSEN, H. (2008).