Studentized Tests of Independence: Random-Lifter approach

TL;DR

This paper introduces the Random-Lifter approach for independence testing, producing simple, normally distributed test statistics that are easier to compute and maintain competitive power compared to existing complex methods.

Contribution

The paper presents a novel Random-Lifter method that yields normally distributed test statistics for independence testing without sample splitting or complex approximations.

Findings

Test statistics follow standard normal distribution under null hypothesis.

Method maintains competitive power with existing tests.

Numerical simulations and real-data analysis validate effectiveness.

Abstract

The exploration of associations between random objects with complex geometric structures has catalyzed the development of various novel statistical tests encompassing distance-based and kernel-based statistics. These methods have various strengths and limitations. One problem is that their test statistics tend to converge to asymptotic null distributions involving second-order Wiener chaos, which are hard to compute and need approximation or permutation techniques that use much computing power to build rejection regions. In this work, we take an entirely different and novel strategy by using the so-called ``Random-Lifter''. This method is engineered to yield test statistics with the standard normal limit under null distributions without the need for sample splitting. In other words, we set our sights on having simple limiting distributions and finding the proper statistics through reverse engineering. We use the Central Limit Theorems (CLTs) for degenerate U-statistics derived from our novel association measures to do this. As a result, the asymptotic distributions of our proposed tests are straightforward to compute. Our test statistics also have the minimax property. We further substantiate that our method maintains competitive power against existing methods with minimal adjustments to constant factors. Both numerical simulations and real-data analysis corroborate the efficacy of the Random-Lifter method.