{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T01:36:34Z","timestamp":1791941794317,"version":"4.3.4"},"reference-count":39,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2024,7,8]],"date-time":"2024-07-08T00:00:00Z","timestamp":1720396800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2210891"],"award-info":[{"award-number":["DMS-2210891"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>With the advance of science and technology, more and more data are collected in the form of functions. A fundamental question for a pair of random functions is to test whether they are independent. This problem becomes quite challenging when the random trajectories are sampled irregularly and sparsely for each subject. In other words, each random function is only sampled at a few time-points, and these time-points vary with subjects. Furthermore, the observed data may contain noise. To the best of our knowledge, there exists no consistent test in the literature to test the independence of sparsely observed functional data. We show in this work that testing pointwise independence simultaneously is feasible. The test statistics are constructed by integrating pointwise distance covariances (Sz\u00e9kely et\u00a0al., 2007) and are shown to converge, at a certain rate, to their corresponding population counterparts, which characterize the simultaneous pointwise independence of two random functions. The performance of the proposed methods is further verified by Monte Carlo simulations and analysis of real data.<\/jats:p>","DOI":"10.1093\/biomet\/asae035","type":"journal-article","created":{"date-parts":[[2024,7,5]],"date-time":"2024-07-05T16:53:14Z","timestamp":1720198394000},"page":"1187-1199","source":"Crossref","is-referenced-by-count":5,"title":["Testing independence for sparse longitudinal data"],"prefix":"10.1093","volume":"111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3348-8877","authenticated-orcid":false,"given":"Changbo","family":"Zhu","sequence":"first","affiliation":[{"name":"Department of Applied and Computational Mathematics and Statistics, University of Notre Dame , 101H Crowley Hall , Notre Dame, Indiana 46556,","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Junwen","family":"Yao","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of California , Davis, One Shields Avenue , Davis, California 95616,","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jane-Ling","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of California , Davis, One Shields Avenue , Davis, California 95616,","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2024,7,8]]},"reference":[{"key":"2025070720014191100_asae035-B1","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.3150\/13-BEJ514","article-title":"A consistent test of independence based on a sign covariance related to Kendall\u2019s tau","volume":"20","author":"Bergsma","year":"2014","journal-title":"Bernoulli"},{"key":"2025070720014191100_asae035-B2","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1093\/biomet\/asz024","article-title":"Nonparametric independence testing via mutual information","volume":"106","author":"Berrett","year":"2019","journal-title":"Biometrika"},{"key":"2025070720014191100_asae035-B3","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1214\/aoms\/1177705055","article-title":"Distribution free tests of independence based on the sample distribution function","volume":"32","author":"Blum","year":"1961","journal-title":"Ann. 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