This paper investigates testing for deviation of a high-dimensional mean vector $\boldsymbolμ$. In contrast to the standard one-sample significance test of the form: $H_0^\texttt{e} : \boldsymbolμ = \boldsymbolμ_0$ versus $H_1^\texttt{e} : \boldsymbolμ \neq \boldsymbolμ_0$, we focus on testing the deviation $H_0 : \|\boldsymbolμ - \boldsymbolμ_0\|_2 \ge d_0$ versus $H_1 : \|\boldsymbolμ - \boldsymbolμ_0\|_2 < d_0$ for a prespecified length $d_0 > 0$. Constructing a valid test statistic for this problem is technically nontrivial. By applying the concept of positive and negative feedback processes from control theory, we propose a test statistic based on a two-armed bandit (TAB) process. The deviation test is also extended to the two-sample setting. Simulation experiments confirm a good performance of the tests in finite samples. Finally, a real data analysis demonstrates the practical significance of the proposed deviation tests.

翻译：本文研究高维均值向量$\boldsymbolμ$的偏差检验问题。与标准单样本显著性检验$H_0^\texttt{e} : \boldsymbolμ = \boldsymbolμ_0$对$H_1^\texttt{e} : \boldsymbolμ \neq \boldsymbolμ_0$不同，我们关注对预设长度$d_0 > 0$的偏差检验：$H_0 : \|\boldsymbolμ - \boldsymbolμ_0\|_2 \ge d_0$对$H_1 : \|\boldsymbolμ - \boldsymbolμ_0\|_2 < d_0$。为该问题构建有效的检验统计量在技术上具有挑战性。通过应用控制理论中的正负反馈过程概念，我们提出了一种基于双臂老虎机（TAB）过程的检验统计量。该偏差检验方法进一步推广至双样本场景。仿真实验证实了所提检验方法在有限样本下的良好性能。最后，通过实际数据分析展示了所提偏差检验方法的实用价值。