An adaptive test procedure for high-dimensional regression coefficients

We propose 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 then 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 values of k. An adaptive omnibus test is constructed via a Cauchy combination over a dyadic grid of k values, 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.