Adaptive L-statistics for high dimensional test problem

In this study, we focus on applying L-statistics to the high-dimensional one-sample location test problem. Intuitively, an L-statistic with $k$ parameters tends to perform optimally when the sparsity level of the alternative hypothesis matches $k$. We begin by deriving the limiting distributions for both L-statistics with fixed parameters and those with diverging parameters. To ensure robustness across varying sparsity levels of alternative hypotheses, we first establish the asymptotic independence between L-statistics with fixed and diverging parameters. Building on this, we propose a Cauchy combination test that integrates L-statistics with different parameters. Both simulation results and real-data applications highlight the advantages of our proposed methods.