Deviation Tests for a High-dimensional Mean

Abstract

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