Gaussian Copula Models for Geostatistical Count Data

Gaussian copula models allow a more direct modeling of the marginal distributions and associa- tion structure of the data, but the modeling of geostatistical count data using copulas is more recent and less studied than that of the Generalized Linear Mixed Models (GLMMs) proposed by Diggle et al. (1998). In this dissertation, we consider Gaussian copula models for geostatistical count data. We first describe a class of random field models for geostatistical count data based on Gaussian copulas and study the properties of these random fields. We contrast the correlation structure of one of these Gaussian copula models with that of a Poisson-Gamma 2 model (De Oliveira, 2013) and show that the former is more flexible than the latter in terms of range of feasible correlation, sensitivity to the mean function and modeling of isotropy.

To fit the model, we investigated the computational efficiency of two existing simulated likeli- hood methods, and proposed a new method based on Markov chain Monte Carlo. We first formu- lated the Gaussian copula models hierarchically when the nugget effect is present. Such formu- lation allows us to adapt a new inferential approach, the data cloning method (Lele et al., 2010), to the Gaussian copula models from its use in the GLMMs. Efficient group updating strategies and Langevin-Hastings algorithms are proposed. We also compared the length and the coverage probabilities of the profile-likelihood confidence intervals and the Wald-type confidence intervals after appropriate reparameterization. Finally, two spatial plug-in prediction methods are discussed. An R package gcKrig was developed to implement all the functions and algorithms discussed in this dissertation using advanced computing techniques, such as Rcpp, RcppArmadillo and parallel computing.