Nonparametric Estimation and Testing for Positive Quadrant Dependent Bivariate Copula

Abstract

In many practical scenarios (e.g., finance, system reliability, etc.), it is often of interest to estimate a bivariate distribution and test for some desired association properties like positive quadrant dependent (PQD) or negative quadrant dependent (NQD). Often estimation and testing for PQD/NQD property are performed using copula models as it then eliminates the need for estimating marginal distributions. Many parametric copula families have been used that allow for controlling the PQD/NQD property by a finite dimensional parameter (often just real-valued) and the problem reduces to the straightforward estimation and testing for fixed dimensional parameter using standard statistical methodologies (e.g., maximum likelihood). This article extends such a line of work by dropping any parametric assumptions and provides a fully data-dependent automated approach to estimate a copula and test for PQD property. The estimator is shown to be large-sample consistent under a set of mild regularity conditions. Numerical illustrations based on simulated data are also provided to compare the performance of the proposed testing procedure with some available methods and applications to real case studies are also provided.

Keywords:

• Anderson–Darling distance
• Bernstein copula
• Positive quadrant dependent

Supplementary Materials

R codes implement copula estimation methods developed in the paper. The goal of R function copula.est is to estimate copula with or without PQD constraints using Bernstein polynomials.

Acknowledgments

The authors are grateful to the two anonymous reviewers and an associate editor for their valuable comments and constructive suggestions, which have led to a significant improvement of an earlier version of this article.