Inverse Norm Weighted Maxsum Test for High Dimensional Location Parameters

Abstract: In the context of high-dimensional data, we investigate the one-sample location testing problem. We introduce a max-type test based on the weighted spatial sign, which exhibits exceptional performance, particularly in the presence of sparse alternatives. Notably, we find that the inverse norm test significantly enhances the power of the test compared to several existing max-type tests. Next, we prove the asymptotic independence between the newly proposed max-type test statistic and the sum-type test statistic based on the weighted spatial sign. Then, we propose an innovative max-sum type testing procedure that integrates both test statistics. This novel procedure demonstrates remarkable robustness and effectiveness across a wide range of signal sparsity levels and heavy-tailed distributions. Through extensive simulation studies, we highlight the superior performance of the proposed method, showcasing its robustness and efficiency compared to traditional alternatives in various high-dimensional settings.