Bartlett Adjustments for Hypothesis Testing in Linear Models with General Error Covariance Matrices

Consider the problem of testing a linear hypothesis of regression coefficients in a general linear regression model with an error term having a covariance matrix involving several nuisance parameters. Three typical test statistics of Wald, Score and Likelihood Ratio (LR) and their Bartlett adjustments have been derived in the literature when the unknown nuisance parameters are estimated by maximum likelihood (ML). On the other hand, statistical inference in linear mixed models has been studied actively and extensively in recent years with applications to smallarea estimation. The marginal distribution of the linear mixed model is included in the framework of the general linear regression model, and the nuisance parameters correspond to the variance components and others in the linear mixed model. Although the restricted ML (REML), minimum norm quadratic unbiased estimator (MINQUE) and other specific estimators are available for estimating the variance components, the Bartlett adjustments given in the literature are not correct for those estimators other than ML. In this paper, using the Taylor series expansion, we derive the Bartlett adjustments of the Wald, Score and modified LR tests for general consistent estimators of the unknown nuisance parameters. These analytical results may be harder to calculate for a model with a complicate structure of the covariance matrix. Thus, we propose the simple parametric bootstrap methods for estimating the Bartlett adjustments and show that they have the second order accuracy. Finally, it is shown that both Bartlett adjustments work well through simulation experiments in the nested error regression model.