[Paper Review] Singularity-agnostic incomplete U-statistics for testing polynomial constraints in Gaussian covariance matrices

Dennis Y.C. Leung, Nils Sturma|arXiv (Cornell University)|Jan 4, 2024

Random Matrices and Applications4 citations

TL;DR

This paper proposes a singularity-agnostic incomplete U-statistic to test polynomial constraints in Gaussian covariance matrices, addressing the slow convergence of the classical Wald test under near-singularity. By leveraging refined Berry-Esseen bounds and sub-Weibull concentration inequalities, the method achieves uniform convergence rates independent of the singularity status of the null hypothesis, ensuring reliable inference even when the gradient of the constraint vanishes.

ABSTRACT

Testing the goodness-of-fit of a model with its defining functional constraints in the parameters could date back to Spearman (1927), who analyzed the famous "tetrad" polynomial in the covariance matrix of the observed variables in a single-factor model. Despite its long history, the Wald test typically employed to operationalize this approach could produce very inaccurate test sizes in many situations, even when the regular conditions for the classical normal asymptotics are met and a very large sample is available. Focusing on testing a polynomial constraint in a Gaussian covariance matrix, we obtained a new understanding of this baffling phenomenon: When the null hypothesis is true but "near-singular", the standardized Wald test exhibits slow weak convergence, owing to the sophisticated dependency structure inherent to the underlying U-statistic that ultimately drives its limiting distribution; this can also be rigorously explained by a key ratio of moments encoded in the Berry-Esseen bound quantifying the normal approximation error involved. As an alternative, we advocate the use of an incomplete U-statistic to mildly tone down the dependence thereof and render the speed of convergence agnostic to the singularity status of the hypothesis. In parallel, we develop a Berry-Esseen bound that is mathematically descriptive of the singularity-agnostic nature of our standardized incomplete U-statistic, using some of the finest exponential-type inequalities in the literature.

Motivation & Objective

• To address the poor finite-sample performance of the classical Wald test when testing polynomial constraints in Gaussian covariance matrices under near-singularity.
• To resolve the slow weak convergence of the Wald test, which arises due to complex dependency structures in U-statistics underlying the test.
• To develop a test statistic whose convergence rate is independent of the singularity status of the null hypothesis.
• To establish a new Berry-Esseen bound that explicitly captures the singularity-agnostic nature of the proposed incomplete U-statistic.
• To provide a theoretically grounded, robust alternative to the Wald test in algebraic statistics and structural equation modeling contexts.

Proposed method

• The paper introduces an incomplete U-statistic that mildly reduces dependence in the original U-statistic by truncating higher-order interactions, thereby stabilizing convergence under near-singularity.
• It employs decoupling inequalities and exponential-type concentration bounds (e.g., sub-Weibull norms) to control tail behavior of polynomial functions of Gaussian variables.
• A novel Berry-Esseen bound is derived using hypercontractivity and moment ratio analysis, explicitly quantifying the normal approximation error in terms of the underlying U-statistic's structure.
• The method relies on a refined moment analysis of degree-2r polynomials in Gaussian variables, leveraging Lemma 2.2 and hypercontractivity to bound higher moments.
• The standardized test statistic is constructed as a ratio of the estimated constraint function to its estimated standard error, using a modified variance estimator based on the incomplete U-statistic.
• Theoretical guarantees are derived via a hierarchical decomposition of indices and maximal inequalities over subsets, ensuring uniform control across all possible singularity configurations.

Figure 1.1. The empirical test sizes (produced by $1000$ repeated experiments) of two types of statistics with critical values calibrated based on their asymptotic null distribution $\mathcal{N}(0,1)$ , plotted against various target nominal levels. These statistics test the particular tetrad $f(\Th

Experimental results

Research questions

• RQ1 Why does the classical Wald test exhibit slow convergence and inaccurate size when testing polynomial constraints in Gaussian covariance matrices under near-singularity?
• RQ2 How can the dependency structure inherent in U-statistics be tamed to achieve uniform convergence rates regardless of the singularity status of the null hypothesis?
• RQ3 Can a Berry-Esseen bound be constructed that explicitly reflects the singularity-agnostic nature of a test statistic in this context?
• RQ4 What role do moment ratios and sub-Weibull norms play in quantifying the convergence speed of U-statistic-based tests under weak regularity conditions?
• RQ5 Is it possible to construct a test statistic whose limiting distribution is well-approximated by normality at a rate independent of whether the null hypothesis is regular, singular, or near-singular?

Key findings

• The classical Wald test suffers from slow weak convergence under near-singularity due to intricate dependency structures in the underlying U-statistic, even when sample size is large and regularity conditions are satisfied.
• The proposed incomplete U-statistic achieves uniform convergence rates across all singularity regimes by reducing the strength of dependence in the U-statistic's kernel.
• A new Berry-Esseen bound is established that explicitly depends on the moment ratio of the U-statistic kernel, quantifying the normal approximation error in a way that is invariant to singularity status.
• The method ensures that the standardized test statistic converges to a standard normal distribution at a rate that is independent of whether the gradient of the constraint vanishes at the true parameter.
• Theoretical analysis confirms that the proposed test maintains accurate size control even in the presence of near-singularity, where the Wald test fails.
• The use of sub-Weibull norms and hypercontractivity enables tight control over higher moments of polynomial functions of Gaussian variables, which is essential for deriving uniform concentration bounds.