{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T16:57:40Z","timestamp":1791392260534,"version":"4.3.3"},"reference-count":47,"publisher":"Institute of Mathematical Statistics","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2025,10,1]]},"abstract":"<jats:p>Functional data analysis is an important research field in statistics which treats data as random functions drawn from some infinite-dimensional functional space, and functional principal component analysis (FPCA) based on eigen-decomposition plays a central role for data reduction and representation. After nearly three decades of research, there remains a key problem unsolved, namely, the perturbation analysis of covariance operator for diverging number of eigencomponents obtained from noisy and discretely observed data. This is fundamental for studying models and methods based on FPCA, while there has not been substantial progress since Hall, M\u00fcller and Wang (Ann. Statist. 34 (2006) 1493\u20131517)\u2019s result for a fixed number of eigenfunction estimates. In this work, we aim to establish a unified theory for this problem, obtaining upper bounds for eigenfunctions with diverging indices in both the L2 and supremum norms, and deriving the asymptotic distributions of eigenvalues for a wide range of sampling schemes. Our results provide insight into the phenomenon when the L2 bound of eigenfunction estimates with diverging indices is minimax optimal as if the curves are fully observed, and reveal the transition of convergence rates from nonparametric to parametric regimes in connection to sparse or dense sampling. We also develop a double truncation technique to handle the uniform convergence of estimated covariance and eigenfunctions. The technical arguments in this work are useful for handling the perturbation series with noisy and discretely observed functional data and can be applied in models or those involving inverse problems based on FPCA as regularization, such as functional linear regression.<\/jats:p>","DOI":"10.1214\/25-aos2531","type":"journal-article","created":{"date-parts":[[2025,11,10]],"date-time":"2025-11-10T08:00:15Z","timestamp":1762761615000},"page":"2103-2127","source":"Crossref","is-referenced-by-count":3,"title":["Theory of functional principal component analysis for discretely observed data"],"prefix":"10.1214","volume":"53","author":[{"given":"Hang","family":"Zhou","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research & School of Data Science and Society, University of North Carolina at Chapel Hill"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dongyi","family":"Wei","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Peking University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fang","family":"Yao","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and Center for Statistical Science, Peking University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"15","unstructured":"<small>Ferraty, F.<\/small> and <small>Vieu, P.<\/small> (2006). <i>Nonparametric Functional Data Analysis<\/i>: <i>Theory and Practice<\/i>. <i>Springer Series in Statistics<\/i>. 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Supplement to \u201cTheory of functional principal component analysis for discretely observed data.\u201d <a href=\"https:\/\/doi.org\/10.1214\/25-AOS2531SUPP\">https:\/\/doi.org\/10.1214\/25-AOS2531SUPP<\/a>","DOI":"10.1214\/25-AOS2531SUPP"}],"container-title":["The Annals of Statistics"],"original-title":[],"deposited":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T16:30:40Z","timestamp":1791390640000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/annals-of-statistics\/volume-53\/issue-5\/Theory-of-functional-principal-component-analysis-for-discretely-observed-data\/10.1214\/25-AOS2531.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,1]]},"references-count":47,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,10,1]]}},"URL":"https:\/\/doi.org\/10.1214\/25-aos2531","relation":{},"ISSN":["0090-5364","2168-8966"],"issn-type":[{"value":"0090-5364","type":"print"},{"value":"2168-8966","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,1]]}}}