A Continuous—Time Distributed Version of Wald’s Sequential Hypothesis Testing Problem

Abstract

This paper discusses a distributed version of Wald’s sequential hypothesis testing problem in the continuous time framework. For sake of concreteness, two decision-makers equipped with their own sensors, are faced with the following hypothesis testing problem: Decide between hypothesis H ₀ and H ₁, where

$$$$

with μ _i ≠ 0 and σ _i ≠ 0, i = 1, 2, non-random; here the noises {W ¹ _t , t ≥ 0} and {W ² _t , t ≥ 0} are independent Brownian motions.

Data is observed continuously and at each instant in time, each decision-maker can either declare one of the hypotheses to be true or continue collecting data. In either case, they base their individual decisions on the data collected by their own sensors up to that time; they do not communicate with each other and so do not share information. The decisions are selected to minimize a joint cost function with two components, the first one capturing the cost for collecting data, and the second assessing the cost for incorrect decisions. This is the simplest problem of its type, for the coupling between the two decision—makers occurs only through the cost structure. This problem was considered first in discrete—time by Teneketzis [6] who showed that the person-by-person optimal strategy was of threshold type for each sensor. Here a similar result is derived by simple and direct arguments based on well—known facts for the single detector problem. Moreover, explicit formulae are derived for this joint cost function when the detector policies are of threshold type, owing to the fact that at the decision times, the likelihood functionals assume one of two threshold values owing to the continuity of the paths of Brownian motion. This is in sharp contrast with the overshoot phenomena that leads in the discrete—time situation to the celebrated Wald approximations. These explicit formulae not only vividly display the cost interaction taking place between the two sensors but readily allow for a reduction of the original problem to a mathematical programming problem in four variables over a simple constraint set.

The work of this author was supported by an ONR Fellowship.

The work of this author was supported partially through ONR Grant NO0014-84-K-0614, partially through NSF Grant ECS-83-51836 and partially through a Grant from GM Laboratories.

The work of this author was supported partially through NSF Grant NSFD CDR -85-00108 and partially through ONR Grant NO0014-83-K-0731