[Paper Review] A Kernel Test for Three-Variable Interactions with Random Processes

Paul K. Rubenstein, Kacper Chwialkowski+1 more|arXiv (Cornell University)|Mar 2, 2016

Statistical Distribution Estimation and Applications20 references3 citations

TL;DR

This paper proposes a wild bootstrap method for testing three-variable interactions in stationary time series using the kernel-based Lancaster interaction measure, overcoming the failure of permutation bootstrap under temporal dependence. The key contribution is a novel, simplified proof technique leveraging Hilbert space central limit theorems, which establishes the validity of the wild bootstrap and enables more efficient asymptotic analysis than traditional V-statistic decompositions.

ABSTRACT

We apply a wild bootstrap method to the Lancaster three-variable interaction measure in order to detect factorisation of the joint distribution on three variables forming a stationary random process, for which the existing permutation bootstrap method fails. As in the i.i.d. case, the Lancaster test is found to outperform existing tests in cases for which two independent variables individually have a weak influence on a third, but that when considered jointly the influence is strong. The main contributions of this paper are twofold: first, we prove that the Lancaster statistic satisfies the conditions required to estimate the quantiles of the null distribution using the wild bootstrap; second, the manner in which this is proved is novel, simpler than existing methods, and can further be applied to other statistics.

Motivation & Objective

• To develop a valid statistical test for three-variable interactions in stationary time series where i.i.d. assumptions fail.
• To address the limitation of permutation bootstrap methods in temporal dependence settings.
• To establish the theoretical validity of the wild bootstrap for the Lancaster three-variable interaction test statistic.
• To provide a simpler, more general proof framework for asymptotic analysis of kernel-based test statistics.

Proposed method

• The paper applies the wild bootstrap to the Lancaster three-variable interaction measure to estimate the null distribution under temporal dependence.
• It proves that the Lancaster test statistic satisfies the conditions required for wild bootstrap validity under β-mixing conditions.
• The method treats the test statistic as the norm of a Hilbert space operator and applies a Central Limit Theorem for Hilbert space-valued random variables.
• A novel proof technique avoids the complex Hoeffding decomposition used in prior V-statistic analyses, reducing algebraic burden.
• The approach is generalized to other kernel statistics, as demonstrated with the Hilbert-Schmidt Independence Criterion (HSIC).
• Theoretical results are supported by showing that lower-order terms decay at rate Op(n^{-1/2}), leaving the dominant asymptotic behavior governed by higher-order kernel components.

Experimental results

Research questions

• RQ1 Can the wild bootstrap be applied to the Lancaster three-variable interaction test under temporal dependence?
• RQ2 Does the Lancaster test statistic satisfy the conditions for wild bootstrap validity in stationary processes?
• RQ3 Can a simpler proof technique replace the traditional Hoeffding decomposition for kernel test statistics?
• RQ4 Is the proposed Hilbert space approach generalizable to other kernel-based dependence measures?
• RQ5 How does the performance of the wild bootstrap Lancaster test compare to existing methods in detecting weak joint but strong individual influences?

Key findings

• The wild bootstrap is valid for the Lancaster three-variable interaction test under β-mixing conditions, providing a correct resampling method where permutation bootstrap fails.
• The proposed proof technique is significantly simpler than existing V-statistic decomposition methods, avoiding factorial growth in terms.
• The method establishes that the test statistic converges in probability to the norm of a population-centered Hilbert space operator.
• Lower-order terms in the test statistic decay at rate Op(n^{-1/2}), confirming their asymptotic irrelevance.
• The approach yields a shorter and more transparent proof for the HSIC test statistic compared to prior work.
• The Lancaster test outperforms existing methods in detecting strong joint influence when individual influences are weak.