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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 algorithm originally designed to recursively fit nonparametric mixture models. The resulting predictive recursion $ e $-process affords anytime-valid inference and is asymptotically efficient in the sense that its growth rate is first-order optimal relative to the predictive recursion\u2019s mixture model.<\/jats:p>","DOI":"10.1093\/biomet\/asae066","type":"journal-article","created":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T15:24:45Z","timestamp":1732721085000},"source":"Crossref","is-referenced-by-count":2,"title":["Anytime-valid and asymptotically efficient inference driven by predictive recursion"],"prefix":"10.1093","volume":"112","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7157-3354","authenticated-orcid":false,"given":"Vaidehi","family":"Dixit","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Missouri , 314 N Middlebush Hall , Columbia, Missouri 65201,","place":["USA"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0760-9145","authenticated-orcid":false,"given":"Ryan","family":"Martin","sequence":"additional","affiliation":[{"name":"Department of Statistics, North Carolina State University , 2311 Stinson Drive , Raleigh, North Carolina 27607,","place":["USA"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2024,11,29]]},"reference":[{"key":"2025112820012304100_asae066-B1","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316870","volume-title":"Bayesian Theory","author":"Bernardo","year":"1994"},{"key":"2025112820012304100_asae066-B2","volume-title":"Elements of Information Theory","author":"Cover","year":"2006","edition":"2nd ed"},{"key":"2025112820012304100_asae066-B3","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1111\/j.1467-9868.2010.00753.x","article-title":"Maximum likelihood estimation of a multi-dimensional log-concave density","volume":"72","author":"Cule","year":"2010","journal-title":"J. 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