Joint limiting laws for high-dimensional independence tests

Testing independence is of significant interest in many important areas of large-scale inference. Using extreme-value form statistics to test against sparse alternatives and using quadratic form statistics to test against dense alternatives are two important testing procedures for high-dimensional independence. However, quadratic form statistics suffer from low power against sparse alternatives, and extreme-value form statistics suffer from low power against dense alternatives with small disturbances and may have size distortions due to its slow convergence. For real-world applications, it is important to derive powerful testing procedures against more general alternatives. Based on intermediate limiting distributions, we derive (model-free) joint limiting laws of extreme-value form and quadratic form statistics, and surprisingly, we prove that they are asymptotically independent. Given such asymptotic independencies, we propose (model-free) testing procedures to boost the power against general alternatives and also retain the correct asymptotic size. Under the high-dimensional setting, we derive the closed-form limiting null distributions, and obtain their explicit rates of uniform convergence. We prove their consistent statistical powers against general alternatives. We demonstrate the performance of our proposed test statistics in simulation studies. Our work provides very helpful insights to high-dimensional independence tests, and fills an important gap.

💡 Research Summary

The paper addresses the problem of testing mutual independence among the components of a high‑dimensional random vector when the dimension p may be comparable to or even exponentially larger than the sample size n. Two classical families of test statistics are considered. The first, a quadratic‑form statistic (S_n=\sum_{i<j}\hat\sigma_{ij}^2), aggregates the squared off‑diagonal entries of the sample covariance matrix and is powerful against dense alternatives where many small correlations are present. The second, an extreme‑value statistic (L_n=\max_{i<j}|\hat\sigma_{ij}|), focuses on the largest absolute sample correlation and is powerful against sparse alternatives where only a few large correlations exist. Each statistic alone suffers from low power against the opposite type of alternative, and the extreme‑value statistic converges slowly to its Gumbel limit, causing size distortions in finite samples.

The authors’ main contribution is to study the joint limiting behavior of these two statistics. By introducing an intermediate limiting distribution for the extreme‑value statistic (denoted (F(y))), they obtain uniform convergence rates for both (S_n) and (L_n). Lemma 1 shows that a properly centered and scaled version of (S_n) converges to the standard normal distribution, with an explicit error bound that depends on the relative growth of p and n. Lemma 2 establishes that the distribution of (L_n) after the usual centering and scaling is uniformly close to (F(y)), and the distance to the final Gumbel limit is of order ((\log p)^3/n).

The pivotal result, Theorem 1, proves that the joint distribution (P_{S_n,L_n}(z,y)) factorizes asymptotically into the product of the marginal distributions, up to an error of order (\min(p^{-1/5},p^{,n/p})). In other words, (S_n) and (L_n) are asymptotically independent despite being constructed from the same set of sample correlations. Theorem 2 refines this by showing that the joint distribution is asymptotically (\Phi(z)F(y)), where (\Phi) is the standard normal cdf. Corollary 1 then translates the result into the familiar limiting law (\Phi(z)