{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,26]],"date-time":"2026-10-26T17:41:42Z","timestamp":1793036502047,"version":"4.5.0"},"reference-count":50,"publisher":"Institute of Mathematical Statistics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2026,2,1]]},"abstract":"<jats:p>Traditional static functional data analysis is facing new challenges due to streaming data, where data constantly flows in. A major challenge is that storing such an ever-increasing amount of data in memory is nearly infeasible. In addition, existing inferential tools in online learning are mainly developed for finite-dimensional problems, while inference methods for functional data are focused on the offline settings. In this paper, we focus on the online estimation of functional linear regression with a scalar response and a functional covariate. To tackle these issues in this setting, we develop a functional stochastic gradient descent algorithm and propose an online bootstrap resampling procedure that enables us to perform the local (and global) inference for the slope function in functional linear regression. In particular, the proposed estimation and inference procedures use only one pass over the data; thus they are easy to implement and suitable to the situation where data arrive in a streaming manner. Furthermore, we establish the convergence rate and the asymptotic distribution for the proposed slope function estimator. The proposed perturbed estimator from the bootstrap procedure is shown to enjoy the same theoretical properties, which provide the theoretical justification for our online inference tool. As far as we know, this is the first inference result on functional linear regression with streaming data. Numerical studies are conducted to demonstrate the finite-sample performance of our proposed procedure.<\/jats:p>","DOI":"10.1214\/25-aos2549","type":"journal-article","created":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T06:27:47Z","timestamp":1772605667000},"page":"120-145","source":"Crossref","is-referenced-by-count":2,"title":["Scalable inference in functional linear regression with streaming data"],"prefix":"10.1214","volume":"54","author":[{"given":"Jinhan","family":"Xie","sequence":"first","affiliation":[{"name":"Yunnan Key Laboratory of Statistical Modeling and Data Analysis, Yunnan University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Enze","family":"Shi","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Statistical Sciences, University of Alberta"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peijun","family":"Sang","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Waterloo"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zuofeng","family":"Shang","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, New Jersey Institute of Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bei","family":"Jiang","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Statistical Sciences, University of Alberta"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3011-9216","authenticated-orcid":false,"given":"Linglong","family":"Kong","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Statistical Sciences, University of Alberta"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"44","doi-asserted-by":"publisher","unstructured":"<small>Yao, F.<\/small>, <small>M\u00fcller, H.-G.<\/small> and <small>Wang, J.-L.<\/small> (2005). 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