One-Sided Tests in Linear Models with Multivariate t-Distribution

Abstract

In this paper, we examine the problem of testing equality and inequality constraints on the regression coefficients in linear models with multivariate t-distribution for the within-subject errors. Iterative processes for evaluating the parameters under equality and inequality constraints are presented for fixed degrees of freedom. Under certain regularity assumptions the likelihood ratio, score (Rao) and Wald tests are asymptotically distributed as a mixture of chi-squared distributions, where the weights do not depend on the null parameters, but may depend on the correlations. Thus, one has to search through the set of correlation coefficients for least favorable points. Some particular cases are discussed and the empirical performances of the statistical tests for small and moderate sample sizes are compared via Monte Carlo studies. Comparisons between the theoretical and empirical distributions of the statistical one-sided tests are also made. The quality of the approximation seems to be good even for small (about n = 20) sample sizes. We present an illustrative example with real data in which statistical tests based on normal and t models are compared, confirming the robustness of the t models against outliers. This robustness is reflected directly on the decision from the one-sided tests.