Power one sequential tests exist for weakly compact $\mathscr P$ against $\mathscr P^c$

2026-04-03

AI Summary

This work investigates the existence of sequential tests with power one for distinguishing a weakly compact composite null hypothesis $\mathscr{P}$ from its complement $\mathscr{P}^c$ in Polish spaces under i.i.d. sampling. By integrating weak compactness, sequential testing theory, and $e$-process methodology, the paper establishes a general sufficient condition and demonstrates that for any weakly compact $\mathscr{P}$, there exists a level-$\alpha$ sequential test achieving power one over $\mathscr{P}^c$. Moreover, it explicitly constructs an $e$-process that diverges to infinity almost surely under every distribution in $\mathscr{P}^c$, and proves that this $e$-process attains optimal asymptotic relative growth rate, thereby substantially strengthening existing results in the literature.

Abstract

Suppose we observe data from a distribution $P$ and we wish to test the composite null hypothesis that $P\in\mathscr P$ against a composite alternative $P\in \mathscr Q\subseteq \mathscr P^c$. Herbert Robbins and coauthors pointed out around 1970 that, while no batch test can have a level $α\in(0,1)$ and power equal to one, sequential tests can be constructed with this fantastic property. Since then, and especially in the last decade, a plethora of sequential tests have been developed for a wide variety of settings. However, the literature has not yet provided a clean and general answer as to when such power-one sequential tests exist. This paper provides a remarkably general sufficient condition (that we also prove is not necessary). Focusing on i.i.d. laws in Polish spaces without any further restriction, we show that there exists a level-$α$ sequential test for any weakly compact $\mathscr P$, that is power-one against $\mathscr P^c$ (or any subset thereof). We show how to aggregate such tests into an $e$-process for $\mathscr P$ that increases to infinity under $\mathscr P^c$. We conclude by building an $e$-process that is asymptotically relatively growth rate optimal against $\mathscr P^c$, an extremely powerful result.

Problem

• sequential test
• power-one
• weakly compact
• composite hypothesis
• e-process

Innovation

• power-one sequential test
• weakly compact
• e-process
• Polish space
• asymptotic growth rate optimality