A class of sequential multi-hypothesis tests

TL;DR

This paper introduces a simplified, generally suboptimal, class of sequential multi-hypothesis tests called DBC, which are easier to implement and show high efficiency compared to optimal and existing tests across various scenarios.

Contribution

It proposes the DBC class of tests that drop backward control, simplifying the construction of sequential tests for multiple hypotheses, and demonstrates their high efficiency through numerical comparisons.

Findings

DBC tests achieve over 99.3% efficiency compared to optimal tests.

DBC tests outperform MSPRT in some scenarios.

Numerical examples confirm high performance of DBC tests.

Abstract

In this paper, we deal with sequential testing of multiple hypotheses. In the general scheme of construction of optimal tests based on the backward induction, we propose a modification which provides a simplified (generally speaking, suboptimal) version of the optimal test, for any particular criterion of optimization. We call this DBC version (the one with Dropped Backward Control) of the optimal test. In particular, for the case of two simple hypotheses, dropping backward control in the Bayesian test produces the classical sequential probability ratio test (SPRT). Similarly, dropping backward control in the modified Kiefer-Weiss solutions produces Lorden's 2-SPRTs . In the case of more than two hypotheses, we obtain in this way new classes of sequential multi-hypothesis tests, and investigate their properties. The efficiency of the DBC-tests is evaluated with respect to the optimal Bayesian multi-hypothesis test and with respect to the matrix sequential probability ratio test (MSPRT) by Armitage. In a multihypothesis variant of the Kiefer-Weiss problem for binomial proportions the performance of the DBC-test is numerically compared with that of the exact solution. In a model of normal observations with a linear trend, the performance of of the DBC-test is numerically compared with that of the MSPRT. Some other numerical examples are presented. In all the cases the proposed tests exhibit a very high efficiency with respect to the optimal tests (more than 99.3\% when sampling from Bernoulli populations) and/or with respect to the MSPRT (even outperforming the latter in some scenarios).