Predictive Inference with Copulas for Bivariate Data

Abstract

Nonparametric predictive inference (NPI) is a statistical approach with strong frequentist properties, with inferences explicitly in terms of one or more future observations. NPI is based on relatively few modelling assumptions, enabled by the use of lower and upper probabilities to quantify uncertainty. While NPI has been developed for a range of data types, and for a variety of applications, thus far it has not been developed for multivariate data. This thesis presents the rst study in this direction. Restricting attention to bivariate data, a novel approach is presented which combines NPI for the marginals with copulas for representing the dependence between the two variables. It turns out that, by using a discretization of the copula, this combined method leads to relatively easy computations. The new method is introduced with use of an assumed parametric copula. The main idea is that NPI on the marginals provides a level of robustness which, for small to medium-sized data sets, allows some level of misspecication of the copula.
As parametric copulas have restrictions with regard to the kind of dependency they can model, we also consider the use of nonparametric copulas in combination with NPI for the marginals. As an example application of our new method, we consider accuracy of diagnostic tests with bivariate outcomes, where the weighted combination of both variables can lead to better diagnostic results than the use of either of the variables alone. The results of simulation studies are presented to provide initial insights into the performance of the new methods presented in this thesis, and examples using data from the literature are used to illustrate applications of the methods. As this is the rst research into developing NPI-based methods for multivariate data, there are many related research opportunities and challenges, which we briefly discuss.