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: $s = \frac{m_{3}}{\sqrt{m_{2}}^{3}}$ 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):

$\beta = \frac{3(n^{2} + 27n - 70)(n + 1)(n + 3)}{(n - 2)(n + 5)(n + 7)(n + 9)}$ [1]

$\omega^{2} = - 1 + \sqrt{2(\beta - 1)}$ [2]

$\delta = \frac{1}{\sqrt{\log{(\omega}^{2})}}$ [3]

$y = s\sqrt{\frac{(\omega^{2} - 1)(n + 1)(n + 3)}{12(n - 2)}}$ [4]

$z_{1} = \delta log(y + \sqrt{y^{2} - 1})$ [5]

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:

$\delta = (n - 3)(n + 1)(n^{2} + 15n - 4)$ [6]

$a = \frac{(n - 2)(n + 5)(n + 7)(n^{2} + 27n - 70)}{6\delta}$ [7]

$c = \frac{(n - 7)(n + 5)(n + 7)(n^{2} + 2n - 5)}{6\delta}$ [8]

$l= \frac{(n + 5)(n + 7)({n^{3} + 37n}^{2} + 11n - 313)}{12\delta}$ [9]

$\alpha = a + c \times s^{2}$ [10]

$\chi = 2l(k - 1 - s^{2})$ [11]

$z_{2} = \sqrt{9\alpha}\left( \frac{1}{9\alpha} - 1 + \sqrt[3]{\frac{\chi}{2\alpha}} \right)$ [12]

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.:

$DH = z_{1}^{2} + z_{2}^{2}\sim\chi^{2}(2)$ [13]