Fitting and evaluating certain two-level hierarchical models

This thesis consists of three papers on fitting and evaluating certain two-level hierarchical models.; The first paper presents and evaluates a new procedure for making inferences about the random effects parameters in a two level Normal hierarchical model. The maximum likelihood estimate gives overly narrow confidence intervals for these individual effects. The ADM (adjustment for density maximization) and a new approximation avoid these problems while retaining the MLE's ease of fitting. The procedure approximates the posterior means and variances corresponding to a superharmonic prior on the between group variance component. The resulting approximations are shown to be close to the exact values. The new rule is shown to have excellent frequentist properties in its own right and it sheds light on the performance of the exact Bayes rule.; The second paper exploits and extends recent developments in the method of data augmentation for dynamic linear models which aim to improve the convergence rate of the EM algorithm while maintaining its attractive convergence properties such as monotone convergence in loglikelihood without adding significantly to the computational complexity of the algorithm. We adapt the one-step-late EM algorithm to PXEM to establish a fast closed form algorithm with monotone convergence for fully Bayesian calculations in a general setting. The various algorithms are illustrated in several examples which demonstrate a computational savings of as much as 99.9%, with the biggest savings occurring when EM algorithm is the slowest to converge.; The third paper presents an independent importance sampling method to get posterior draws of parameters for a two-level univariate Gamma hierarchical regression model. This sampling approach is efficient (with RNE > 85%) in all examples here. We then evaluate the frequentist properties of these inference rules and show substantial gain, in terms of coverage probabilities and of quadratic risks in comparison with the coverage and risks of commonly used usual plug-in estimates and with others. This method can be applied to model sample variances and reliability. We point out the equivalence of this Gamma model with a Poisson hierarchical model.