{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T05:31:20Z","timestamp":1778650280997,"version":"3.51.4"},"reference-count":37,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2024,9,30]],"date-time":"2024-09-30T00:00:00Z","timestamp":1727654400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,1,16]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We address the problem of testing the conditional mean and conditional variance for nonstationary data. We build e-values and p-values for four types of nonparametric composite hypothesis with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality and their combination. Using the obtained e-values and p-values, we construct tests via e-processes, also known as testing by betting, as well as some tests based on combining p-values for comparison. Although we mainly focus on one-sided tests, the two-sided test for the mean is also studied. Simulation and empirical studies are conducted under a few settings, and they illustrate features of the methods based on e-processes.<\/jats:p>","DOI":"10.1093\/biomet\/asae049","type":"journal-article","created":{"date-parts":[[2024,9,30]],"date-time":"2024-09-30T15:40:12Z","timestamp":1727710812000},"source":"Crossref","is-referenced-by-count":3,"title":["Testing the mean and variance by <i>e<\/i>-processes"],"prefix":"10.1093","volume":"112","author":[{"given":"Yixuan","family":"Fan","sequence":"first","affiliation":[{"name":"Centre for Applied Mathematics, Tianjin University , 92 Weijin Road , Tianjin 300072, , yxfanjlu@gmail.com","place":["China"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2410-490X","authenticated-orcid":false,"given":"Zhanyi","family":"Jiao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Illinois State University , 300 S. School Street , Normal, Illinois 61761, U.S.A"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3849-4555","authenticated-orcid":false,"given":"Ruodu","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Waterloo , 200 University Avenue West , Waterloo, Ontario N2L 3G1, , wang@uwaterloo.ca","place":["Canada"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2024,9,30]]},"reference":[{"key":"2025012017232240100_asae049-B1","author":"Ahn","year":"2023"},{"key":"2025012017232240100_asae049-B2","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1111\/j.2517-6161.1995.tb02031.x","article-title":"Controlling the false discovery rate: a practical and powerful approach to multiple testing","volume":"57","author":"Benjamini","year":"1995","journal-title":"J. R. Statist. Soc. 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