{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T03:12:49Z","timestamp":1790651569280,"version":"4.1.0"},"reference-count":53,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2023,6,24]],"date-time":"2023-06-24T00:00:00Z","timestamp":1687564800000},"content-version":"vor","delay-in-days":1,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["EP\/W014971\/1"],"award-info":[{"award-number":["EP\/W014971\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["EP\/V007556\/1"],"award-info":[{"award-number":["EP\/V007556\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["EP\/X002195\/1"],"award-info":[{"award-number":["EP\/X002195\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["CIF-2102227"],"award-info":[{"award-number":["CIF-2102227"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"NIH","doi-asserted-by":"publisher","award":["R01AG061303"],"award-info":[{"award-number":["R01AG061303"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"NIH","doi-asserted-by":"publisher","award":["R01AG062542"],"award-info":[{"award-number":["R01AG062542"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,9,29]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>The Markov property is widely imposed in analysis of time series data. Correspondingly, testing the Markov property, and relatedly, inferring the order of a Markov model, are of paramount importance. In this article, we propose a nonparametric test for the Markov property in high-dimensional time series via deep conditional generative learning. We also apply the test sequentially to determine the order of the Markov model. We show that the test controls the type-I error asymptotically, and has the power approaching one. Our proposal makes novel contributions in several ways. We utilise and extend state-of-the-art deep generative learning to estimate the conditional density functions, and establish a sharp upper bound on the approximation error of the estimators. We derive a doubly robust test statistic, which employs a nonparametric estimation but achieves a parametric convergence rate. We further adopt sample splitting and cross-fitting to minimise the conditions required to ensure the consistency of the test. We demonstrate the efficacy of the test through both simulations and the three data applications.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkad064","type":"journal-article","created":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T19:31:32Z","timestamp":1685647892000},"page":"1204-1222","source":"Crossref","is-referenced-by-count":11,"title":["Testing for the Markov property in time series via deep conditional generative learning"],"prefix":"10.1093","volume":"85","author":[{"given":"Yunzhe","family":"Zhou","sequence":"first","affiliation":[{"name":"Division of Biostatistics, University of California at Berkeley , Berkeley, CA , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chengchun","family":"Shi","sequence":"additional","affiliation":[{"name":"London School of Economics and Political Science , London , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2962-1989","authenticated-orcid":false,"given":"Lexin","family":"Li","sequence":"additional","affiliation":[{"name":"Division of Biostatistics, University of California at Berkeley , Berkeley, CA , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2065-8486","authenticated-orcid":false,"given":"Qiwei","family":"Yao","sequence":"additional","affiliation":[{"name":"London School of Economics and Political Science , London , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,6,23]]},"reference":[{"key":"2023101106481728900_qkad064-B1","author":"A\u00eft-Sahalia","year":"1997"},{"key":"2023101106481728900_qkad064-B2","doi-asserted-by":"crossref","DOI":"10.1093\/biomet\/77.4.669","article-title":"Identification of nonlinear time series: First order characterization and order determination","volume":"77","author":"Auestad","year":"1990,","journal-title":"Biometrika"},{"issue":"3","key":"2023101106481728900_qkad064-B3","doi-asserted-by":"crossref","first-page":"930","DOI":"10.1109\/18.256500","article-title":"Universal approximation bounds for superpositions of a sigmoidal function","volume":"39","author":"Barron","year":"1993","journal-title":"IEEE Transactions on Information Theory"},{"key":"2023101106481728900_qkad064-B4","author":"Belloni","year":"2018"},{"key":"2023101106481728900_qkad064-B5","author":"Bishop","year":"1994,"},{"key":"2023101106481728900_qkad064-B6","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1214\/154957805100000104","article-title":"Basic properties of strong mixing conditions. 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