On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds

Abstract

We present two general lower bounds for stopping times of sequential tests between arbitrary composite nulls $\mathcal P$ and alternatives $\mathcal Q$. The first lower bound is for the ``Wald setting'' where the type-1 error level $\alpha$ approaches zero for a fixed alternative $Q \in \mathcal Q$, and equals $\log(1/\alpha)$ divided by a certain infimum KL divergence between $\mathcal P$ and $Q$, termed $\operatorname{KL_{inf}}$. The second lower bound applies to the ``Farrell setting'', where $\alpha$ is fixed and $\operatorname{KL_{inf}}$ approaches $0$ along a sequence of alternatives such that the required expected sample size along that sequence is of order at least $\operatorname{KL^{-1}_{inf}} \log \log \operatorname{KL^{-1}_{inf}}$. Our main contribution is the generality of these bounds, which hold in non-parametric, composite settings, without requiring a dominating reference measure, substantially generalizing the known parametric results. We also provide sufficient conditions for matching upper bounds and show that these are met in several nontrivial non-parametric cases.