Multi-scale tests of independence powerful for detecting explicit or implicit functional relationship

In this article, we consider the problem of testing the independence between two random variables. Our primary objective is to develop tests that are highly effective at detecting associations arising from explicit or implicit functional relationship between two variables. We adopt a multi-scale approach by analyzing neighborhoods of varying sizes within the dataset and aggregating the results. We introduce a general testing framework designed to enhance the power of existing independence tests to achieve our objective. Additionally, we propose a novel test method that is powerful as well as computationally efficient. The performance of these tests is compared with existing methods using various simulated datasets.