Anytime-valid and asymptotically efficient inference driven by predictive recursion.

This article focuses on the construction and theoretical properties of a novel statistical tool called the predictive recursion |$ e $|-process (|$ \small{\rm PR}e $| -process) for distinguishing between two classes of models, particularly in complex nonparametric settings. The |$ \small{\rm PR}e $|-process leverages the predictive recursion algorithm to efficiently fit nonparametric mixture models, enabling anytime-valid inference that controls error rates regardless of stopping rules. The paper establishes that the |$ \small{\rm PR}e $|-process is a valid |$ e $|-process with asymptotically optimal growth rates under alternatives, supported by theoretical results and numerical examples including tests for monotonicity, parametric null models, log-concavity, and time homogeneity. The approach offers computational advantages over existing methods due to its recursive updating and flexibility, while motivating further research on convergence rates and high-dimensional extensions.