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Deviation Tests for a High-dimensional Mean

Zengjing Chen, Ruihan Liu, Jianfeng Yao

Abstract

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.

Deviation Tests for a High-dimensional Mean

Abstract

This paper investigates testing for deviation of a high-dimensional mean vector . In contrast to the standard one-sample significance test of the form: versus , we focus on testing the deviation versus for a prespecified length . 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.
Paper Structure (8 sections, 4 theorems, 87 equations, 2 figures, 2 tables)

This paper contains 8 sections, 4 theorems, 87 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. One-sample case
  3. Two-sample case
  4. Numerical experiments
  5. Empirical study
  6. Appendix
  7. Auxiliary lemmas
  8. Proof of Theorem 1

Key Result

Theorem 2.1

Under Assumptions Ap of high dimensionality and Ap of noise, let us with $\tau_1:=\Vert\boldsymbol{\mu}\Vert_2^2-d_0^2$ and $\sigma_1^2:=\boldsymbol{\mu}'\boldsymbol{\Sigma}\boldsymbol{\mu}+\frac{1}{T_1}\operatorname{Tr}(\boldsymbol{\Sigma}^2)$, we have as $T_2\to\infty$

Figures (2)

  • Figure 1: Density plot of bandit distributions under different values of $\kappa$. The continuous curve ($\kappa=0$) is the standard normal distribution.
  • Figure 2: Density plot of $\mathcal{M}_{T_2,T_2}(\vec{\theta}_{T_2})\overset{d}{\sim}\mathcal{B}(-\kappa_{1,T_2})$ under $H_1$, where the red area is the rejection region $|\mathcal{M}_{T_2,T_2}(\vec{\theta}_{T_2})|>z_{\alpha/2}$. In case 1 as $|\kappa_{1,T_2}|\to\infty$, the blue dashed line suggests the densities of $\mathcal{M}_{T_2,T_2}(\vec{\theta}_{T_2})$ almost lie in the rejection region, so the asymptotic power is full. In case 2 as $|\kappa_{1,T_2}|\to O(1)$, the purple dotted line shows that only part of $\mathcal{M}_{T_2,T_2}(\vec{\theta}_{T_2})$'s distributions lies in the rejection region, so the asymptotic power is in $(\alpha,1)$. In case 3 as $|\kappa_{1,T_2}|\to0$, $\mathcal{M}_{T_2,T_2}(\vec{\theta}_{T_2})$ is asymptotically normal (continuous curve) and the asymptotic power equals $\alpha$.

Theorems & Definitions (7)

  • Remark 2.1
  • Theorem 2.1
  • Theorem 3.1
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof