HZmvntest

Henze and Zirkler (1990) introduce a multivariate version of the univariate There are many tests for assessing the multivariate normality in the statistical literature (Mecklin and Mundfrom, 2003). Unfortunately, there is no known uniformly most powerful test and it is recommended to perform several test to assess it. It has been found that the Henze and Zirkler test have a good overall power against alternatives to normality.

The Henze-Zirkler test is based on a nonnegative functional distance that measures the distance between two distribution functions: the characteristic function of the multivariate normality and the empirical characteristic function.

The Henze-Zirkler statistic is approximately distributed as a lognormal. The lognormal distribution is used to compute the null hypothesis probability.

According to Henze-Wagner (1997), this test has the desirable properties of,

--affine invariance
--consistency against each fixed nonnormal alternative distribution
--asymptotic power against contiguous alternatives of order n^-1/2
--feasibility for any dimension and any sample size

If the data is multivariate normality, the test statistic HZ is approximately lognormally distributed. It proceeds to calculate the mean, variance and smoothness parameter. Then, mean and variance are lognormalized and the P-value is estimated.

Also, for all the interested people, we provide the lognormal critical value.

Inputs:
X - data matrix (size of matrix must be n-by-p; data=rows,
independent variable=columns)
c - covariance normalized by n (=1, default)) or n-1 (~=1)
alpha - significance level (default = 0.05)

Output:
- Henze-Zirkler's Multivariate Normality Test