Deviation Tests for a High-dimensional Mean

This paper investigates testing for deviation of a high-dimensional mean vector $\boldsymbol{\mu}$. In contrast to the standard one-sample significance test of the form: $H_0^\texttt{e} : \boldsymbol{\mu} = \boldsymbol{\mu}_0$ versus $H_1^\texttt{e} : \boldsymbol{\mu} \neq \boldsymbol{\mu}_0$, we focus on testing the deviation $H_0 : \|\boldsymbol{\mu} - \boldsymbol{\mu}_0\|_2 \ge d_0$ versus $H_1 : \|\boldsymbol{\mu} - \boldsymbol{\mu}_0\|_2 < d_0$ 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.