The Effect of Skewness and Kurtosis on Mean and Covariance Structure Analysis: The Univariate Case and Its Multivariate Implication

Abstract

The maximum likelihood (ML) method, based on the normal distribution assumption, is widely used in mean and covariance structure analysis. With typical nonnormal data, the ML method will lead to biased statistics and inappropriate scientific conclusions. This article develops a simple but informative case to show how ML results are influenced by skewness and kurtosis. Specifically, the authors discuss how skewness and kurtosis in a univariate distribution affect the standard errors of the ML estimators, the covariances between the estimators, and the likelihood ratio test of hypotheses on mean and variance parameters. They also describe corrections that have been developed to allow appropriate inference. Enough details are provided so that this material can be used in graduate instruction. For each result, the corresponding results in the higher dimensional case are pointed out, and references are provided.