{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T18:01:07Z","timestamp":1791568867742,"version":"4.3.4"},"reference-count":27,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T00:00:00Z","timestamp":1664323200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11931001"],"award-info":[{"award-number":["11931001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871080"],"award-info":[{"award-number":["11871080"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"publisher","award":["2020YFE0204200"],"award-info":[{"award-number":["2020YFE0204200"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,5,15]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>Despite extensive studies on functional linear regression, there exists a fundamental gap in theory between the ideal estimation from fully observed covariate functions and the reality that one can only observe functional covariates discretely with noise. The challenge arises when deriving a sharp perturbation bound for the estimated eigenfunctions in the latter case, which renders existing techniques for functional linear regression not applicable. We use a pooling method to attain the estimated eigenfunctions and propose a sample-splitting strategy to estimate the principal component scores, which facilitates the theoretical treatment for discretely observed data. The slope function is estimated by approximated least squares, and we show that the resulting estimator attains the optimal convergence rates for both estimation and prediction when the number of measurements per subject reaches a certain magnitude of the sample size. This phase transition phenomenon differs from the known results for the pooled mean and covariance estimation, and reveals the elevated difficulty in estimating the regression function. Numerical experiments, using simulated and real data examples, yield favourable results when compared with existing methods.<\/jats:p>","DOI":"10.1093\/biomet\/asac053","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T05:05:42Z","timestamp":1664427942000},"page":"381-393","source":"Crossref","is-referenced-by-count":28,"title":["Functional linear regression for discretely observed data: from ideal to reality"],"prefix":"10.1093","volume":"110","author":[{"given":"Hang","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences Department of Probability and Statistics, , Center for Statistical Science, Peking University, Beijing 100871, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fang","family":"Yao","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences Department of Probability and Statistics, , Center for Statistical Science, Peking University, Beijing 100871, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huiming","family":"Zhang","sequence":"additional","affiliation":[{"name":"University of Macau Department of Mathematics, , Avenida da Universidade, Macau 999078, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2022,9,28]]},"reference":[{"key":"2023051508572716500_B1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1154-9","volume-title":"Linear Processes in Function Spaces","author":"Bosq,","year":"2000"},{"key":"2023051508572716500_B2","doi-asserted-by":"crossref","first-page":"2159","DOI":"10.1214\/009053606000000830","article-title":"Prediction in functional linear regression","volume":"34","author":"Cai,","year":"2006","journal-title":"Ann. 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