Correlated multivariate Poisson and binary variables occur naturally in medical, biological and epidemiological longitudinal studies. Modeling and simulating such variables is difficult because the correlations are restricted by the marginal means via Fréchet bounds in a complicated way. In this dissertation we will first discuss partially specified models and methods for estimating the regression and correlation parameters. We derive the asymptotic distributions of these parameter estimates. Using simulations based on extensions of the algorithm due to Sim (1993, Journal of Statistical Computation and Simulation, 47, pp. 1–10), we study the performance of these estimates using infeasibility, coverage probabilities of the confidence ellipsoids, and asymptotic relative efficiencies as the criteria.
The second part of this dissertation is devoted to the study of fully specified models constructed using copulas, with special emphasis on the normal copula. Finding the maximum likelihood estimates and the Fisher information matrix for these models requires computation of multivariate normal probabilities. We also discuss several efficient algorithms for calculating multivariate normal integrals. For the multivariate probit and multivariate Poisson log-normal models, we implement maximum likelihood, derive the necessary equations, and illustrate it on two real life data sets. Next we study over and under dispersed models including quasi-multinomial and Lagrange families of distributions. We implement the maximum likelihood method for the quasi-multinomial model and illustrate the application of this model for market analysis of household preferences for saltine crackers
