Copula Based Hierarchical Bayesian Models

The main objective of our study is to employ copula methodology to develop Bayesian hierarchical models to study the dependencies exhibited by temporal, spatial and spatio-temporal processes. We develop hierarchical models for both discrete and continuous outcomes. In doing so we expect to address the dearth of copula based Bayesian hierarchical models to study hydro-meteorological events and other physical processes yielding discrete responses. First, we present Bayesian methods of analysis for longitudinal binary outcomes using Generalized Linear Mixed models (GLMM). We allow flexible marginal association among the repeated outcomes from different time-points. An unique property of this copula-based GLMM is that if the marginal link function is integrated over the distribution of the random effects, its form remains same as that of the conditional link function. This unique property enables us to retain the physical interpretation of the fixed effects under conditional and marginal model and yield proper posterior distribution. We illustrate the performance of the posited model using real life AIDS data and demonstrate its superiority over the traditional Gaussian random effects model. We develop a semiparametric extension of our GLMM and re-analyze the data from the AIDS study. Next, we propose a general class of models to handle non-Gaussian spatial data. The proposed model can deal with geostatistical data that can accommodate skewness, tail-heaviness, multimodality. We fix the distribution of the marginal processes and induce dependence via copulas. We illustrate the superior predictive performance of our approach in modeling precipitation data as compared to other kriging variants. Thereafter, we employ mixture kernels as the copula function to accommodate non-stationary data. We demonstrate the adequacy of this non-stationary model by analyzing permeability data. In both cases we perform extensive simulation studies to investigate the performances of the posited models under misspecification. Finally, we take up the important problem of modeling multivariate extreme values with copulas. We describe, in detail, how dependences can be induced in the block maxima approach and peak over threshold approach by an extreme value copula. We prove the ability of the posited model to handle both strong and weak extremal dependence and derive the conditions for posterior propriety. We analyze the extreme precipitation events in the continental United States for the past 98 years and come up with a suite of predictive maps.