Inference on variance components near boundary in linear mixed effect models

In making inference on variance components in linear mixed effect models, variance component parameters may be located on some boundary of a constrained parameter space, and hence usual asymptotic theory on parameter estimation, test statistics, and information criteria may not hold. We illustrate such boundary issues on variance components, and introduce some methodologies and properties along with literature. The maximum likelihood estimator of the variance parameter vector near some boundary distributes asymptotically as a projection of a normal random vector onto the boundary. The null distribution of the likelihood ratio test statistic is complicated, and hence it has been studied both from asymptotic and numerical aspects. Moreover, a boundary issue in model selection using information criteria is also essential and is closely related to that on the likelihood ratio test. We also describe a boundary issue on testing for linearity of a regression function using the relationship between linear mixed effect models and penalized spline models. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Data Reduction, Smoothing, and Filtering Statistical Models > Model Selection Statistical and Graphical Methods of Data Analysis > Bootstrap and Resampling.