UC Riverside

Historically, the collection of available statistical models for fitting data has been, more or less, restricted to those which are analytically tractable. However, computing power today permits us to use models that, while more complex and often devoid of closed-form distribution or density functions, provide better fits to data. In this thesis, statistical theory, primarily parameter estimation, is developed for three such models: Arnold & Ng's bivariate beta family and its Arnold & Ghosh subfamily of copulas, an 8-parameter family of bivariate Asymmetric Laplace distributions, and a collection of compound random variables. All are distributions whose densities, in general, cannot be written down, but whose realizations can easily be generated via simulation. I apply, and adapt, several methods of likelihood-free statistical inference; including Modified Maximum Likelihood Estimation (MMLE), Approximate Bayesian Computation (ABC), and Markov-Chain Monte-Carlo (MCMC); to achieve various forms of parameter estimates.

For each of the models studied, sub-models were identified and cataloged. In doing so, care is taken to assure that a reasonable balance between the dimensionality of the parameter spaces and the flexibility of the resulting models is maintained. Moreover, in one case, a collection of sub-models is formed, each of which permits simple parameter estimation, while one of the models from the collection is chosen to provide the best fit according to a particular metric. This is done to simplify parameter estimation through dimension reduction while maintaining a high level of diversity of available models. In other cases, a more direct approach is taken, where the model is selected based on prior knowledge.