Asymptotics of Sequential Composite Hypothesis Testing under Probabilistic Constraints

TL;DR

This paper analyzes the fundamental limits of sequential composite hypothesis testing under probabilistic constraints, deriving error exponents and a strong converse, with technical innovations in second-order asymptotics.

Contribution

It provides the first characterization of first- and second-order error exponents for constrained sequential composite testing with convex parameter sets.

Findings

Derived the set of all first-order error exponents.

Proved a strong converse for the testing problem.

Established second-order error exponents under finite alphabet assumptions.

Abstract

We consider the sequential composite binary hypothesis testing problem in which one of the hypotheses is governed by a single distribution while the other is governed by a family of distributions whose parameters belong to a known set $\Gamma$. We would like to design a test to decide which hypothesis is in effect. Under the constraints that the probabilities that the length of the test, a stopping time, exceeds $n$ are bounded by a certain threshold $\epsilon$, we obtain certain fundamental limits on the asymptotic behavior of the sequential test as $n$ tends to infinity. Assuming that $\Gamma$ is a convex and compact set, we obtain the set of all first-order error exponents for the problem. We also prove a strong converse. Additionally, we obtain the set of second-order error exponents under the assumption that $\mathcal{X}$ is a finite alphabet. In the proof of second-order asymptotics, a main technical contribution is the derivation of a central limit-type result for a maximum of an uncountable set of log-likelihood ratios under suitable conditions. This result may be of independent interest. We also show that some important statistical models satisfy the conditions.