Hypothesis Testing of One-Sample Mean Vector in Distributed Frameworks

TL;DR

This paper develops distributed hypothesis tests for the mean vector in large-scale data settings, balancing communication costs and statistical power, and extends classical tests to distributed frameworks for both low and high dimensions.

Contribution

It introduces novel distributed test statistics for mean vector hypotheses, reducing communication costs while analyzing the power tradeoffs compared to centralized tests.

Findings

Distributed tests significantly reduce communication costs.

Tradeoff exists between test power and communication efficiency.

Numerical results validate theoretical insights.

Abstract

Distributed frameworks are widely used to handle massive data, where sample size $n$ is very large, and data are often stored in $k$ different machines. For a random vector $X\in \mathbb{R}^p$ with expectation $\mu$, testing the mean vector $H_0: \mu=\mu_0$ vs $H_1: \mu\ne \mu_0$ for a given vector $\mu_0$ is a basic problem in statistics. The centralized test statistics require heavy communication costs, which can be a burden when $p$ or $k$ is large. To reduce the communication cost, distributed test statistics are proposed in this paper for this problem based on the divide and conquer technique, a commonly used approach for distributed statistical inference. Specifically, we extend two commonly used centralized test statistics to the distributed ones, that apply to low and high dimensional cases, respectively. Comparing the power of centralized test statistics and the distributed ones, it is observed that there is a fundamental tradeoff between communication costs and the powers of the tests. This is quite different from the application of the divide and conquer technique in many other problems such as estimation, where the associated distributed statistics can be as good as the centralized ones. Numerical results confirm the theoretical findings.