Copula bounds for circular data

TL;DR

This paper extends classical copula bounds to circular data, redefining dependency measures for periodic variables and demonstrating their behavior through simulations.

Contribution

It introduces modified Fréchet-Hoeffding bounds and copulas specifically for circular data, addressing the challenge of defining dependency in periodic variables.

Findings

Redefinition of copula bounds for circular data

Development of modified Fréchet and Mardia copulas for circular variables

Simulation results illustrating the behavior of the proposed models

Abstract

We propose the extension of Fr\'{e}chet-Hoeffding copula bounds for circular data. The copula is a powerful tool for describing the dependency of random variables. In two dimensions, the Fr\'{e}chet-Hoeffding upper (lower) bound indicates the perfect positive (negative) dependence between two random variables. However, for circular random variables, the usual concept of dependency is not accepted because of their periodicity. In this work, we redefine Fr\'{e}chet-Hoeffding bounds and consider modified Fr\'{e}chet and Mardia families of copulas for modelling the dependency of two circular random variables. Simulation studies are also given to demonstrate the behavior of the model.