Optimal sequential tests yield log-optimal e-processes

TL;DR

This paper demonstrates that asymptotically optimal sequential tests can be combined into asymptotically log-optimal e-processes using a new class of WAIT e-processes, completing a theoretical framework.

Contribution

It proves the converse that optimal tests can be aggregated into log-optimal e-processes, advancing the understanding of sequential testing theory.

Findings

Asymptotically optimal tests can be aggregated into log-optimal e-processes.

Introduces a new class of WAIT e-processes based on weighted aggregates.

Discusses nuances in definitions of asymptotic optimality.

Abstract

It has been recently shown that e-processes are sufficient for sequential testing in the following sense: every level-$\alpha$ sequential test can be obtained by thresholding an e-process at $1/\alpha$. 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.