Testing for Positive Quadrant Dependence

Abstract

We develop an empirical likelihood (EL) approach to test independence of two univariate random variables X and Y versus the alternative that X and Y are strictly positive quadrant dependent (PQD). Establishing this type of ordering between X and Y is of interest in many applications, including finance, insurance, engineering, and other areas. Adopting the framework in Einmahl and McKeague, we create a distribution-free test statistic that integrates a localized EL ratio test statistic with respect to the empirical joint distribution of X and Y. When compared to well-known existing tests and distance-based tests we develop by using copula functions, simulation results show the EL testing procedure performs well in a variety of scenarios when X and Y are strictly PQD. We use three datasets for illustration and provide an online R resource practitioners can use to implement the methods in this article. Supplementary materials for this article are available online.