[Paper Review] Asymptotic efficiency of p-mean tests for means in high dimensions

Iosif Pinelis|arXiv (Cornell University)|Jun 2, 2010

Mathematical Inequalities and Applications8 references3 citations

TL;DR

This paper analyzes the asymptotic relative efficiency (ARE) of p-mean tests for multivariate mean vectors in high-dimensional settings, showing that for p > 2, these tests can significantly outperform the standard likelihood ratio test (LRT) while never being substantially worse. The key result is that AREp,2 for p = 3 ranges from approximately 0.96 to infinity depending on the direction of the alternative mean vector, under general and natural conditions on the asymptotic behavior of sample size and mean vector magnitude.

ABSTRACT

The asymptotic efficiency, ARE_{p,2}, of the tests for multivariate means theta in \R^d based on the p-means relative to the standard 2-mean, (approximate) likelihood ratio test (LRT), is considered for large dimensions d. It turns out that these p-mean tests for p>2 may greatly outperform the LRT while never being significantly worse than the LRT. For instance, ARE_{p,2} for p=3 varies from about 0.96 to \infty, depending on the direction of the alternative mean vector theta_1, for the null hypothesis H_0: theta=\0. These results are based on a complete characterization, under certain general and natural conditions, of the varying pairs (n,theta_1) for which the values of the power of the p-mean test for theta=\0 and theta=theta_1 tend, respectively, to prescribed values alpha and beta. The proofs use such classic results as the Berry-Esseen bound in the central limit theorem and the conditions of convergence to a given infinitely divisible distribution, as well as a recent result by the author on the Schur^2-concavity properties of Gaussian measures.

Motivation & Objective

• To investigate the asymptotic relative efficiency (ARE) of p-mean tests compared to the standard 2-mean (LRT) in high-dimensional multivariate mean testing.
• To characterize the conditions under which p-mean tests achieve desired power levels for large dimensions d.
• To determine when p-mean tests for p > 2 can significantly outperform the LRT while remaining robust in performance.

Proposed method

• Uses a complete characterization of asymptotically sufficient (AS) pairs and triples (n, θ₁) for which the power of the p-mean test converges to prescribed levels α and β under H₀ and H₁.
• Applies classic tools such as the Berry-Esseen bound in the central limit theorem and convergence conditions to infinitely divisible distributions.
• Leverages a recent result on Schur²-concavity properties of Gaussian measures to analyze the behavior of p-mean statistics.
• Analyzes the p-mean of the sample mean vector defined as (1/d ∑|xⱼ|ᵖ)¹ᐟᵖ for p ∈ [−∞, ∞], including limits for p = 0 (geometric mean), p = ∞ (maximum), and p = −∞ (minimum).
• Derives the asymptotic relative efficiency (AREp,2) as the ratio of the sample sizes required by the p-mean test and LRT to achieve the same power for a given significance level.
• Considers the group Gd of coordinate permutations and sign flips, and studies Gd-invariant tests based on p-mean statistics.

Experimental results

Research questions

• RQ1 For which directions of the alternative mean vector θ₁ does the p-mean test with p > 2 achieve higher asymptotic relative efficiency than the LRT?
• RQ2 What is the range of asymptotic relative efficiency (AREp,2) for p-mean tests with p > 2 in high-dimensional settings?
• RQ3 Under what conditions do p-mean tests maintain high power across different directions of the mean vector while being robust to directionality?
• RQ4 How does the choice of p affect the efficiency of the test relative to the LRT, particularly when the true mean vector has only a few dominant components?
• RQ5 Can p-mean tests be uniformly more powerful than the LRT in high dimensions, especially when the mean vector is sparse?

Key findings

• For p = 3, the asymptotic relative efficiency (AREp,2) of the p-mean test relative to the LRT varies from approximately 0.96 to infinity, depending on the direction of the alternative mean vector θ₁.
• The p-mean test for p > 2 can greatly outperform the LRT in high-dimensional settings, particularly when the true mean vector has only a few dominant components.
• The p-mean test never performs significantly worse than the LRT, ensuring robustness across different mean vector configurations.
• The asymptotic relative efficiency is fully characterized under general and natural conditions on the growth of sample size n and the norm of the alternative mean vector θ₁.
• The analysis confirms that p-mean tests are asymptotically sufficient and that their power behavior is governed by majorization and stochastic ordering principles.
• The results are derived using advanced tools including the Berry-Esseen bound, convergence to infinitely divisible laws, and Schur²-concavity of Gaussian measures.