Optimal sequential tests yield log-optimal e-processes

2026-05-12

AI Summary

This work addresses the problem of constructing asymptotically log-optimal e-processes from asymptotically optimal sequential tests. To this end, it introduces a novel class of WAIT e-processes (Weighted Average of Stopped Indicator Tests), which for the first time enables the reverse construction of asymptotically log-optimal e-processes from asymptotically optimal sequential tests. The proposed framework not only establishes a bidirectional equivalence between these two notions of optimality—thereby completing a key theoretical gap—but also clarifies subtle distinctions among different definitions of asymptotic optimality in the literature. The resulting WAIT e-processes are shown to grow to infinity at the optimal rate under the alternative hypothesis, thereby verifying their log-optimality.

Abstract

It has been recently shown that e-processes are sufficient for sequential testing in the following sense: every level-$α$ sequential test can be obtained by thresholding an e-process at $1/α$. However, in the above result, neither does the test have to be asymptotically optimal (in terms of stopping times) nor does the e-process have to be asymptotically log-optimal. It has separately been shown that asymptotically log-optimal e-processes yield asymptotically optimal sequential tests. In this paper, we prove the converse, arguably completing the story: it is possible to aggregate asymptotically optimal sequential tests into asymptotically log-optimal e-processes. This is accomplished by using a new class of WAIT e-processes: those that are Weighted Aggregates of Indicators of stopping Times that begin at zero, are nondecreasing and increase to infinity under the alternative at the optimal rate. Importantly, the paper discusses several nuances in the varied definitions of asymptotic (log-)optimality.

Problem

• sequential testing
• e-processes
• asymptotic optimality
• log-optimality
• stopping times

Innovation

• e-processes
• sequential testing
• asymptotic log-optimality
• WAIT e-processes
• stopping times