🤖 AI Summary
This paper addresses the problem of interpretable detection of dependency structures among random variables. We propose a novel framework integrating a rank-transform-based estimator for the quantile dependence function with a local acceptance region. The method constructs robust quantile dependence measures via rank standardization and employs local hypothesis testing to enable visual diagnostic assessment of dependency patterns and rigorous independence testing under finite samples. Key contributions include: (1) the first nonparametric estimation and theoretical derivation of the quantile dependence function; (2) guaranteed validity of statistical tests at any sample size, balancing high global power with precise localization of heterogeneous dependencies; (3) superior empirical power across diverse alternative models and successful identification of heterogeneous non-independence in real-world data; and (4) a computationally efficient algorithm supporting intuitive, graphical diagnostic interpretation.
📝 Abstract
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.