A General Test for Independent and Identically Distributed Hypothesis

Abstract

We propose a simple and intuitive test for arguably the most prevailing hypothesis in statistics that data are independent and identically distributed (IID), based on a newly introduced off-diagonal sequential U-process. This IID test is fully nonparametric and applicable to random objects in general spaces, while requiring no specific alternatives such as structural breaks or serial dependence, which allows for detecting general types of violations of the IID assumption. An easy-to-implement jackknife multiplier bootstrap is tailored to produce critical values of the test. Under mild conditions, we establish Gaussian approximation for the proposed U-processes, and derive non-asymptotic coupling and Kolmogorov distance bounds for its maximum and the bootstrapped version, providing rigorous theoretical guarantees. Simulations and real data applications are conducted to demonstrate the usefulness and versatility compared with existing methods.