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。与标准单样本显著性检验形式$H_0^\texttt{e} : \boldsymbolμ = \boldsymbolμ_0$ 对比 $H_1^\texttt{e} : \boldsymbolμ \neq \boldsymbolμ_0$不同，我们聚焦于检验如下假设：对于预设长度$d_0 > 0$，以$H_0 : \|\boldsymbolμ - \boldsymbolμ_0\|_2 \ge d_0$ 对比 $H_1 : \|\boldsymbolμ - \boldsymbolμ_0\|_2 < d_0$。为此问题构建有效检验统计量在技术层面具有相当难度。通过引入控制理论中的正负反馈过程概念，我们提出一种基于双臂赌博机（TAB）过程的检验统计量。该偏差检验方法进一步扩展至双样本设定。仿真实验证实了该检验在有限样本中的良好表现。最后，真实数据分析展示了所提偏差检验方法的实际应用价值。