Anytime valid and asymptotically optimal inference driven by predictive recursion

Distinguishing two candidate models is a fundamental and practically important statistical problem. Error rate control is crucial to the testing logic but, in complex nonparametric settings, can be difficult to achieve, especially when the stopping rule that determines the data collection process is not available. This paper proposes an e-process construction based on the predictive recursion (PR) algorithm originally designed to recursively fit nonparametric mixture models. The resulting PRe-process affords anytime valid inference and is asymptotically efficient in the sense that its growth rate is first-order optimal relative to PR's mixture model.