Predictive inference for bivariate data: Combining nonparametric predictive inference for marginals with an estimated copula

Abstract

This paper presents a new method for prediction of an event involving a future bivariate observation. The method combines nonparametric predictive inference (NPI) applied to the marginals with a parametric copula to model and estimate the dependence structure between two random quantities, as such the method is semi-parametric. In NPI, uncertainty is quantified through imprecise probabilities. The resulting imprecision in the marginals provides robustness with regard to the assumed parametric copula. Due to the specific nature of NPI, the estimation of the copula parameter is also quite straightforward. The performance of this method is investigated via simulations, with particular attention to robustness with regard to the assumed copula in case of small data sets. The method is further illustrated via two examples, using small data sets from the literature. This paper presents several novel aspects of statistical inference. First, the link between NPI and copulas is powerful and attractive with regard to computation. Secondly, statistical methods using imprecise probability have gained substantial attention in recent years, where typically imprecision is used on aspects for which less information is available. This paper presents a different approach, namely imprecision mainly being introduced on the marginals, for which there is typically quite sufficient information, in order to provide robustness for the harder part of the inference, namely the copula assumptions and estimation. Thirdly, the set up of the simulations to evaluate the performance of the proposed method is novel, key to these are frequentist comparisons of the success proportion of predictions with the corresponding data-based lower and upper predictive inferences. All these novel ideas can be applied far more generally to other inferences and models, while also many alternatives can be considered. Hence, this paper presents the starting point of an extensive research programme towards powerful predictive inference methods for multi-variate data.