{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T01:58:17Z","timestamp":1790819897978,"version":"4.1.0"},"reference-count":33,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T00:00:00Z","timestamp":1678665600000},"content-version":"vor","delay-in-days":3686,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-0805975"],"award-info":[{"award-number":["DMS-0805975"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01-NS060910"],"award-info":[{"award-number":["R01-NS060910"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000097","name":"National Center for Research Resources","doi-asserted-by":"publisher","award":["UL1-RR024996"],"award-info":[{"award-number":["UL1-RR024996"]}],"id":[{"id":"10.13039\/100000097","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11201390"],"award-info":[{"award-number":["11201390"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2013,6,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>We propose a fast penalized spline method for bivariate smoothing. Univariate P-spline smoothers are applied simultaneously along both co-ordinates. The new smoother has a sandwich form which suggested the name \u2018sandwich smoother\u2019 to a referee. The sandwich smoother has a tensor product structure that simplifies an asymptotic analysis and it can be fast computed. We derive a local central limit theorem for the sandwich smoother, with simple expressions for the asymptotic bias and variance, by showing that the sandwich smoother is asymptotically equivalent to a bivariate kernel regression estimator with a product kernel. As far as we are aware, this is the first central limit theorem for a bivariate spline estimator of any type. Our simulation study shows that the sandwich smoother is orders of magnitude faster to compute than other bivariate spline smoothers, even when the latter are computed by using a fast generalized linear array model algorithm, and comparable with them in terms of mean integrated squared errors. We extend the sandwich smoother to array data of higher dimensions, where a generalized linear array model algorithm improves the computational speed of the sandwich smoother. One important application of the sandwich smoother is to estimate covariance functions in functional data analysis. In this application, our numerical results show that the sandwich smoother is orders of magnitude faster than local linear regression. The speed of the sandwich formula is important because functional data sets are becoming quite large.<\/jats:p>","DOI":"10.1111\/rssb.12007","type":"journal-article","created":{"date-parts":[[2013,2,7]],"date-time":"2013-02-07T13:27:17Z","timestamp":1360243637000},"page":"577-599","source":"Crossref","is-referenced-by-count":90,"title":["Fast Bivariate<i>P<\/i>-Splines: The Sandwich Smoother"],"prefix":"10.1093","volume":"75","author":[{"given":"Luo","family":"Xiao","sequence":"first","affiliation":[{"name":"Johns Hopkins University , Baltimore , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yingxing","family":"Li","sequence":"additional","affiliation":[{"name":"Xiamen University , People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Ruppert","sequence":"additional","affiliation":[{"name":"Cornell University , Ithaca , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2013,2,7]]},"reference":[{"key":"2023031400060236000_","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1093\/biomet\/asp035","article-title":"Asymptotic properties of penalized spline estimators","volume":"96","author":"Claeskens","year":"2009","journal-title":"Biometrika"},{"key":"2023031400060236000_","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1111\/j.1467-9868.2006.00543.x","article-title":"Generalized linear array models with applications to multidimensional smoothing","volume":"68","author":"Currie","year":"2006","journal-title":"J. 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