MISSPECIFIED MODELS WITH PARAMETERS OF INCREASING DIMENSION

We study a special class of misspecified generalized linear 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 both in linear models and generalized linear models under which the MLE derived from the misspecified working model converges to a well defined limit, and is asymptotically normal. The sample variance under the linear model is biased under model misspecification; but there exists a robust variance estimator of the MLE that converges to the true variance in probability. Criterion-based automatic model selection methods may select a linear model that contains many extra variables, but this can be avoided by using the robust variance estimator for the MLE $\hat{\bbeta}_n$ in Bonferroni-adjusted model selection and by choosing $\lambda_n$ that grows fast enough in Shao's GIC. 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. We find that when the link function in generalized linear mixed models is correctly specified, the estimated parameters have entries that are close to zero except for those corresponding to the fixed effects in the true model. The estimated variance of the MLE is always smaller (in computational examples) than the true variance of the MLE, but the robust ``sandwich'' variance estimator can estimate the true variance very well, and extra significant variables will appear only when the link function is not correctly specified.