[Paper Review] Asymptotically Independent U-Statistics in High-Dimensional Testing

Yinqiu He, Gongjun Xu+2 more|arXiv (Cornell University)|Sep 2, 2018

Random Matrices and Applications66 references4 citations

TL;DR

This paper proposes a family of asymptotically independent U-statistics for high-dimensional global testing, enabling adaptive combination of p-values from different orders to achieve robust power across sparse, dense, and intermediate alternatives. The key contribution is establishing asymptotic independence and normality of U-statistics of different orders under the null, allowing valid p-value combination via Fisher's method with superior power across diverse alternatives.

ABSTRACT

Many high-dimensional hypothesis tests aim to globally examine marginal or low-dimensional features of a high-dimensional joint distribution, such as testing of mean vectors, covariance matrices and regression coefficients. This paper constructs a family of U-statistics as unbiased estimators of the $\ell_p$-norms of those features. We show that under the null hypothesis, the U-statistics of different finite orders are asymptotically independent and normally distributed. Moreover, they are also asymptotically independent with the maximum-type test statistic, whose limiting distribution is an extreme value distribution. Based on the asymptotic independence property, we propose an adaptive testing procedure which combines $p$-values computed from the U-statistics of different orders. We further establish power analysis results and show that the proposed adaptive procedure maintains high power against various alternatives.

Motivation & Objective

• Address the challenge of low statistical power in high-dimensional global testing when the true alternative structure (sparse, dense, or intermediate) is unknown.
• Develop a unified framework for constructing unbiased estimators of ℓp-norms of high-dimensional parameters (e.g., mean vectors, covariance matrices, regression coefficients).
• Establish asymptotic independence and normality of U-statistics of different orders under the null hypothesis to enable valid p-value combination.
• Propose an adaptive testing procedure that combines p-values from multiple U-statistics to maintain high power across diverse alternative types.
• Provide theoretical power analysis and numerical validation showing robust performance across a wide range of alternatives.

Proposed method

• Construct U-statistics of order a as unbiased estimators of ∥E∥a_a = ∑_{l∈L} e^a_l using kernel functions of i.i.d. observations.
• Define the U-statistic of order a as U(a) = (P^n_a)^{-1} ∑_{l∈L} ∑_{1≤i1≠⋯≠ia≤n} ∏_{k=1}^a K_l(z_{i_{(k-1)γ_l+1}}, ..., z_{i_{kγ_l}}), where K_l is an unbiased kernel for parameter e_l.
• Leverage the asymptotic independence of U-statistics of different orders under the null hypothesis, proven via joint asymptotic normality.
• Combine p-values from U-statistics of multiple orders using Fisher’s method to form an adaptive test with robust power.
• Use permutation or asymptotic approximations for the limiting distribution of the maximum-type statistic (U(∞)) under the null.
• Apply the adaptive procedure to real-world problems such as two-sample mean and covariance testing, and GWAS applications.

Experimental results

Research questions

• RQ1 Can U-statistics of different orders be asymptotically independent under the high-dimensional null hypothesis?
• RQ2 Does combining p-values from U-statistics of multiple orders yield a testing procedure with high power across diverse alternative types (sparse, dense, intermediate)?
• RQ3 How do the proposed U-statistics perform in comparison to existing sum-of-squares and maximum-type tests under faint, sparse, and moderate alternatives?
• RQ4 What is the theoretical justification for the asymptotic independence of U-statistics of different orders and their limiting distributions?
• RQ5 Can the adaptive procedure maintain high power when the true alternative structure is unknown or varies across settings?

Key findings

• U-statistics of different finite orders are asymptotically independent and jointly normally distributed under the null hypothesis, enabling valid p-value combination.
• The U-statistic of order a=2 is a sum-of-squares-type estimator, while higher orders (e.g., a→∞) approximate the maximum-type statistic, capturing different alternative structures.
• In the extreme faint case (Model 1), U(1) and U(2) achieve empirical power of 0.416 and 0.484 at p=300, respectively, with the adaptive procedure (adpUf2) reaching 0.781.
• In the extreme sparse case (Model 2), higher-order U-statistics (U(4), U(∞)) achieve near-1 power, while U(1) and U(2) drop to 0.049 and 0.122 at p=300.
• In the moderately sparse and faint case (Model 3), finite-order U(5) is most powerful (0.946 at p=300), and the adaptive procedure (adpUf2) maintains high power (0.940 at p=300) where other methods like Chen, Sriva, and Schott fail.
• The proposed adaptive procedure (adpUf2) achieves comparable or better power than computationally expensive methods (Pan1, Pan2) while being significantly more efficient, especially at larger p.