Some New Test Statistics for Mean and CovarianceStructure Analysis with High Dimensional

Covariance structure analysis is often used for inference and for dimension reduction with high dimensional data. When data is not normally distributed, the asymptotic distribution free (ADF) method is often used to t a proposed model. This approach uses a weight matrix based on the inverse of the matrix formed by the sample fourth moments and sample covariances. The ADF test statistic is asymptotically distributed as a chi-square variate, but its empirical performance rejects the true model too often at all but impractically large sample sizes. By comparing mean and covariance structure analysis with its peer in the multivariate linear model, we propose some modi ed ADF test statistics as F-tests whose distributions we approximate using F-distributions. Empirical studies show that the distributions of the new F-tests are more closely approximated by F-distributions than are the original ADF statistics when referred to chi-square distributions. Detailed analysis indicates why the ADF statistic fails on large models. An explanation for the improved behavior of Yuan and Bentler's statistic is also given. Implications for power analysis and model tests in other areas are discussed.