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.