Title: Copula by triangulation
Authors: Yajing Zhu - Concordia University (Canada) [presenting]
Artem Prokhorov - University of Sydney (Australia)
Edward Anderson - University of Sydney (Australia)
Abstract: The focus is on simple arrangements for approximating copula densities with spline type surfaces while guaranteeing that our estimator is indeed a copula density. We start by showing the difficulty of approximating copula densities with piecewise linear surface while guaranteeing the uniform margin property because this implies that the estimation procedure would involve mixed integer optimization problems. We then turn to a straightforward method of applying the spline as basis functions to approximating copula densities. We propose a semi-parametric copula density estimation procedure that guarantees that the estimator is indeed a copula density. The estimation procedure involves a maximum likelihood estimation of the coefficients of the splines. With simple linear constraints included in the maximization problem, we are solving a convex optimization problem which is easy to solve numerically. Our estimation procedure can be easily generalized to irregular grid on the unit square instead of regular grid with equidistant knots, which implies good localization property. Our estimator also can be easily generated to higher dimensions. We construct a simulation-based study to examine the effect of sample sizes and extent of dependence on the performance of our copula density estimation method and compare with the leading copula density estimators.