Missing the Random Effect

Revision with unchanged content. In this project we look at the case when two of fundamental assumptions in the method of Maximum Likelihood are violated. In particular, we study a special class of misspecified models, where the true model is a mixed effect model but the working model is a fixed effect model with parameters of dimension increasing with sample size. We provide a sufficient condition under which the MLE derived from the working model converges to a welldefined and asymptotically normally-distributed limit. In linear models, the sample variance is biased; but there exists a robust variance estimator of the MLE that converges to the true variance in probability. We also study the Criterion-based automatic model selection methods and find that they may select a linear model that contains spurious variables, but this can be avoided by using the robust variance estimator for the MLE in Bonferroni-adjusted model section or by choosing λn that grows fast enough in Shao's GIC. In generalized linear models, general results are given and computational and simulation studies are carried out to corroborate asymptotic theoretical results as well as to calculate quantities that are not available in theoretical calculation.