Improving the (approximate) sequential probability ratio test by avoiding overshoot

The sequential probability ratio test (SPRT) by Wald (1945) is a cornerstone of sequential analysis. Based on desired type-I, II error levels $\alpha, \beta \in (0,1)$, it stops when the likelihood ratio statistic crosses certain upper and lower thresholds, guaranteeing optimality of the expected sample size. However, these thresholds are not closed form and the test is often applied with approximate thresholds $(1-\beta)/\alpha$ and $\beta/(1-\alpha)$ (approximate SPRT). When $\beta > 0$, this neither guarantees type I,II error control at $\alpha,\beta$ nor optimality. When $\beta=0$ (power-one SPRT), it guarantees type I error control at $\alpha$ that is in general conservative, and thus not optimal. The looseness in both cases is caused by overshoot: the test statistic overshoots the thresholds at the stopping time. One standard way to address this is to calculate the right thresholds numerically, but many papers and software packages do not do this. In this paper, we describe a different way to improve the approximate SPRT: we change the test statistic to avoid overshoot. Our technique uniformly improves power-one SPRTs $(\beta=0)$ for simple nulls and alternatives, or for one-sided nulls and alternatives in exponential families. When $\beta > 0$, our techniques provide valid type I error guarantees, lead to similar type II error as Wald's, but often needs less samples. These improved sequential tests can also be used for deriving tighter parametric confidence sequences, and can be extended to nontrivial settings like sampling without replacement and conformal martingales.