  Sequential tests and their implied confidence sequences, which are valid at
arbitrary stopping times, promise flexible statistical inference and on-the-fly
decision making. However, strong guarantees are limited to parametric
sequential tests that under-cover in practice or concentration-bound-based
sequences that over-cover and have suboptimal rejection times. In this work, we
consider classic delayed-start normal-mixture sequential probability ratio
tests, and we provide the first asymptotic type-I-error and
expected-rejection-time guarantees under general non-parametric data generating
processes, where the asymptotics are indexed by the test's burn-in time. The
type-I-error results primarily leverage a martingale strong invariance
principle and establish that these tests (and their implied confidence
sequences) have type-I error rates asymptotically equivalent to the desired
(possibly varying) $\alpha$-level. The expected-rejection-time results
primarily leverage an identity inspired by It\^o's lemma and imply that, in
certain asymptotic regimes, the expected rejection time is asymptotically
equivalent to the minimum possible among $\alpha$-level tests. We show how to
apply our results to sequential inference on parameters defined by estimating
equations, such as average treatment effects. Together, our results establish
these (ostensibly parametric) tests as general-purpose, non-parametric, and
near-optimal. We illustrate this via numerical simulations and a real-data
application to A/B testing at Netflix.
