THE COMBINATION TEST FOR MULTIVARIATE NORMALITY

A basic graphical approach for checking normality is the Q-Q plot that compares sample quantiles against the population quantiles. In the univariate analysis, the probability plot correlation coefficient test for normality has been studied extensively. We consider testing the multivariate normality by using the correlation coefficient of the Q-Q plot. When multivariate normality holds, the sample squared distance should follow a chi-square distribution for large samples. The plot should resemble a straight line. A correlation coefficient test can be constructed by using the pairs of points in the probability plot. When the correlation coefficient test does not reject the null hypothesis, the sample data may come from a multivariate normal distribution or some other distributions. So, we use the following two steps to test multivariate normality. First, we check the multivariate normality by using the probability plot correction coefficient test. If the test does not reject the null hypothesis, then we test symmetry of the distribution and determine whether multivariate normality holds. This test procedure is called the combination test. The size and power of this test are studied, and it is found that the combination test, in general, is more powerful than other tests for multivariate normality.