Adaptive Test Procedure for High Dimensional Regression Coefficient

We develop a unified $L$-statistic testing framework for high-dimensional regression coefficients that adapts to unknown sparsity. The proposed statistics rank coordinate-wise evidence measures and aggregate the top $k$ signals, bridging classical max-type and sum-type tests. We establish joint weak convergence of the extreme-value component and standardized $L$-statistics under mild conditions, yielding an asymptotic independence that justifies combining multiple $k$’s. An adaptive omnibus test is constructed via a Cauchy combination over a dyadic grid of $k$, and a wild bootstrap calibration is provided with theoretical guarantees. Simulations demonstrate accurate size and strong power across sparse and dense alternatives, including non-Gaussian designs.

💡 Research Summary

The paper addresses the global testing problem in high‑dimensional linear regression, where the number of covariates can far exceed the sample size. The authors focus on testing the null hypothesis that a high‑dimensional block of regression coefficients β_b is identically zero, possibly after adjusting for a low‑dimensional nuisance block β_a. Classical approaches fall into two families: max‑type (ℓ∞) tests that are powerful for sparse alternatives and sum‑type (ℓ2) or energy tests that excel for dense alternatives. Because the sparsity level of the true signal is unknown in practice, a single test tuned to one regime can suffer substantial power loss when the alternative deviates from that regime.

To bridge this gap, the authors introduce a family of L‑statistics indexed by a truncation parameter k. For each coordinate j they compute a standardized marginal score W_j = (X̃_jᵀ ε̂ / σ̂‖X̃_j‖)², where X̃_j denotes the residualized design after projecting out the nuisance covariates. The scores are sorted in decreasing order, W_(1) ≥ … ≥ W_(m), and the top‑k statistic is defined as L_k = Σ_{j=1}^k W_(j). When k = 1, L_k reduces to a max‑type statistic; when k = m, it becomes an ℓ2‑type sum. Intermediate values of k interpolate between the two extremes, allowing the procedure to adapt to unknown sparsity.

The theoretical contribution is twofold. First, under Gaussian design and mild dependence conditions (Assumptions A1–A4), the authors derive the joint weak limit of the ordered scores for fixed k. The extreme‑value component converges to a Poisson point process after centering by b_m = 2 log m − log log m, yielding a Gumbel‑type limit for the maximum. Consequently, the distribution of L_k for fixed k can be expressed via a convolution of extreme‑value and Gaussian components. Second, when k grows proportionally to the dimension (k = ⌈γ m⌉ with γ∈(0,1)), they prove a central limit theorem for L_k. The limiting mean μ_γ and variance σ_γ² involve threshold‑dependent moments of a χ²₁ variable and the covariance structure Σ_{b|a} of the residualized covariates. Importantly, Theorem 3 establishes that the extreme‑value part (e.g., the top one or ten scores) and the standardized L_{⌈γ m⌉} are asymptotically independent. This independence justifies combining p‑values from multiple k’s without inflating type‑I error.

For practical implementation, a wild bootstrap with Rademacher multipliers is employed. Bootstrap responses Y_i* = X_{i,a} β̂_a + ε̂_i ξ_i (ξ_i ∈ {±1}) are generated under the null, and the entire L‑statistic family is recomputed for each bootstrap replicate. Monte‑Carlo p‑values p_k = (1/B)∑_{b=1}^B 1{L_k^{*(b)} ≥ L_k} are obtained for any k, providing a unified calibration for both fixed‑k and diverging‑k statistics.

To achieve adaptivity, the authors select a dyadic grid of truncation levels k_i = ⌊2^{-i} m⌋ for i = 1,…,K, where K ≈ log₂(m/20). For each k_i they compute the bootstrap p‑value p_{k_i}, and they also compute p‑values for two extreme‑value components (e.g., the top score and the 10th largest score). These p‑values are combined using the Cauchy combination test: T_C = Σ tan