The Type-II Error of Test Supermartingales: e-Power versus the Chernoff-Stein Exponent

Abstract

In safe hypothesis testing with test supermartingales, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $\alpha \in (0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $1/\alpha$. Due to an inherent asymmetry, the type-II error behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate ${\mathbb{E}}_{P_1}[\log E]$, the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level $c>0$, every $\alpha$ and horizon $t$ we construct e-variables of conditional e-power exactly $c$ whose probability of not rejecting by $t$ is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, $\Lambda (E)={\sup }_{s\ge 0}\{-\log {\mathbb{E}}_{P_1}[E^{-s}]\}$, whose range is exactly determined: ${\sup }_E\Lambda (E)=\mathrm{KL}(P_0\Vert P_1)$, the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, $\mathrm{KL}(P_1\Vert P_0)$, and that one is attained, $P_0$-a.s. uniquely, by the likelihood ratio $R$. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family $R^{\beta }/{\mathbb{E}}_{P_0}[R^{\beta }]$: the ceiling as $\beta \downarrow 0$, $R$ at $\beta =1$. Which $\beta$ is best is settled by the horizon, exactly: $R$ is optimal at $t=\log (1/\alpha )/\mathrm{KL}(P_1\Vert P_0)$ alone, beaten by sharpening $(\beta >1)$ below it and by flattening above.