On a distribution form of subcopulas

Abstract

© 2020 Elsevier Inc. In this work, we study the problem of (sub)copula estimation via continuity of the Sklar's correspondence. One benefit of this approach is that the estimator can be obtained from that of the corresponding (joint) distribution function via plug-in method. Additional proof is not required. Our approach is to naturally embed the space of subcopulas into the space of distribution functions. This allows us to consider two common modes of convergence, namely, the uniform convergence and the weak convergence on the space of (sub)copulas. Since these modes of convergence are well-studied, we will be able to combine our results with existing results in the same way that is done for distribution functions. Instead of proving these results directly, however, we will apply two common distances characterizing these modes of convergence, namely, the Chebyshev distance and the Levy distance. By using distances, the rate of convergence is also known. Topological properties of the space of subcopulas based on these two distances are also compared and contrasted with that of previously defined distances.