In this paper, we investigate sphericity testing in high-dimensional settings, where existing methods primarily rely on sum-type test procedures that often underperform under sparse alternatives. To address this limitation, we propose two max-type test procedures utilizing the sample covariance matrix and the sample spatial-sign covariance matrix, respectively. Furthermore, we introduce two Cauchy combination test procedures that integrate both sum-type and max-type tests, demonstrating their superiority across a wide range of sparsity levels in the alternative hypothesis. Our simulation studies corroborate these findings, highlighting the enhanced performance of our proposed methodologies in high-dimensional sphericity testi

翻译：本文研究高维背景下的球形性检验问题，现有方法主要依赖于求和型检验程序，其在稀疏备择假设下往往表现不佳。为克服这一局限，我们分别利用样本协方差矩阵与样本空间符号协方差矩阵，提出了两种极大值型检验程序。进一步地，我们引入了两种柯西组合检验程序，将求和型与极大值型检验相结合，证明了其在备择假设多种稀疏度水平下均具有优越性。模拟研究验证了上述结论，凸显了本文所提方法在高维球形性检验中的增强性能。