Detecting dependence structure: visualization and inference

Identifying dependency between two random variables is a fundamental problem. The clear interpretability and ability of a procedure to provide information on the form of possible dependence is particularly important when exploring dependencies. In this paper, we introduce a novel method that employs a new estimator of the quantile dependence function and pertinent local acceptance regions. This leads to an insightful visualisation and a rigorous evaluation of the underlying dependence structure. We also propose a test of independence of two random variables, pertinent to this new estimator. Our procedures are based on ranks, and we derive a finite-sample theory that guarantees the inferential validity of our solutions at any given sample size. The procedures are simple to implement and computationally efficient. The large sample consistency of the proposed test is also proved. We show that, in terms of power, the new test is one of the best statistics for independence testing when considering a wide range of alternative models. Finally, we demonstrate the use of our approach to visualise dependence structure and to detect local departures from independence through analysing some real-world datasets.