A General Test for Independent and Identically Distributed Hypothesis

TL;DR

This paper introduces a fully nonparametric, general test for the IID hypothesis using a novel off-diagonal sequential U-process, with theoretical guarantees and practical validation through simulations and real data.

Contribution

It presents a new, simple, and versatile nonparametric IID test based on off-diagonal sequential U-processes, applicable to general spaces without specific alternative assumptions.

Findings

The test effectively detects violations of IID in simulations.

The bootstrap method provides accurate critical values.

The approach outperforms existing methods in various applications.

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.