Abstract: Several test statistics for covariance structure models derived from the normal theory likelihood ratio are studied. These statistics are robust to certain violations of the multivariate normality assumption underlying the classical method. In order to explicitly model the behavior of these statistics, two new classes of nonnormal distributions are defined and their fourth-order moment matrices are obtained. These nonnormal distributions can be used as alternatives to elliptical symmetric distributions in the study of the robustness of a multivariate statistical method. Conditions for the validity of the statistics under the two classes of nonnormal distributions are given. Some commonly used models are considered as examples to verify our conditions under each class of nonnormal distributions. It is shown that these statistics are valid under much wider classes of distributions than previously assumed. The theory also provides an explanation for previously reported Monte-Carlo results on some of the statistics.
Key words and phrases: Covariance structure, kurtosis, likelihood ratio test, nonnormal distribution, scaled statistics, skewness.