Computes Mardia’s multivariate measures of skewness and kurtosis and tests for multivariate normality.

The type double function is imsls_d_multivar_normality_test.

A pointer to an array of dimension 13 containing output statistics

Function imsls_f_multivar_normality_test computes Mardia’s (1970) measures b1,p and b2,p of multivariate skewness and kurtosis, respectfully, for p = n_variables. These measures are then used in computing tests for multivariate normality. Three test statistics, one based upon b1,p alone, one based upon b2,p alone, and an omnibus test statistic formed by combining normal scores obtained from b1,p and b2,p are computed. On the order of np3, operations are required in computing b1,p when the method of Isogai (1983) is used, where n = n_observations. On the order of np2, operations are required in computing b2,p.

Let

where

fi is the frequency of the i-th observation, and wi is the weight for this observation. (Weights wi are defined such that xi is distributed according to a multivariate normal, N(μ, Σ/wi) distribution, where Σ is the covariance matrix.) Mardia’s multivariate skewness statistic is defined as:

while Mardia’s kurtosis is given as:

Both measures are invariant under the affine (matrix) transformation AX + D, and reduce to the univariate measures when p = n_variables = 1. Using formulas given in Mardia and Foster (1983), the approximate expected value, asymptotic standard error, and asymptotic p‑value for b2,p, and the approximate expected value, an asymptotic chi-squared statistic, and p‑value for the b1,p statistic are computed. These statistics are all computed under the null hypothesis of a multivariate normal distribution. In addition, standard normal scores W1(b1,p) and W2(b2,p) (different from but similar to the asymptotic normal and chi-squared statistics above) are computed. These scores are combined into an asymptotic chi-squared statistic with two degrees of freedom:

This chi-squared statistic may be used to test for multivariate normality. A p‑value for the chi-squared statistic is also computed.

In this example, 150 observations from a 5 dimensional standard normal distribution are generated via routine imsls_f_random_normal (Chapter 12, Random Number Generation). The skewness and kurtosis statistics are then computed for these observations.