Decentralized Two-Sided Sequential Tests for A Normal Mean

TL;DR

This paper develops a decentralized sequential testing method for a normal mean with limited memory sensors, introducing tandem quantizers and invariance principles, achieving asymptotic Bayes optimality and analyzing quantizer choices.

Contribution

It proposes a novel decentralized sequential test using simple quantizers that is asymptotically Bayes, and explores invariant tests with stationary quantizers for normal mean hypotheses.

Findings

The proposed test is asymptotically Bayes optimal.

Quantizer of the form I(X ≥ λ) suffices for optimality.

λ = 0.5 is suboptimal for invariant tests.

Abstract

This article is concerned with decentralized sequential testing of a normal mean $\mu$ with two-sided alternatives. It is assumed that in a single-sensor network system with limited local memory, i.i.d. normal raw observations are observed at the local sensor, and quantized into binary messages that are sent to the fusion center, which makes a final decision between the null hypothesis $H_0: \mu = 0$ and the alternative hypothesis $H_1: \mu = \pm 1.$ We propose a decentralized sequential test using the idea of tandem quantizers (or equivalently, a one-shot feedback). Surprisingly, our proposed test only uses the quantizers of the form $I(X_{n} \ge \lambda),$ but it is shown to be asymptotically Bayes. Moreover, by adopting the principle of invariance, we also investigate decentralized invariant tests with the stationary quantizers of the form $I(|X_{n}| > \lambda),$ and show that $\lambda = 0.5$ only leads to a suboptimal decentralized invariant sequential test. Numerical simulations are conducted to support our arguments.