{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T14:07:52Z","timestamp":1785420472215,"version":"3.56.0"},"reference-count":46,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2020,5,7]],"date-time":"2020-05-07T00:00:00Z","timestamp":1588809600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100008982","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100008982","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,9,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>We consider the problem of approximating smoothing spline estimators in a nonparametric regression model. When applied to a sample of size $n$, the smoothing spline estimator can be expressed as a linear combination of $n$ basis functions, requiring $O(n^3)$ computational time when the number $d$ of predictors is two or more. Such a sizeable computational cost hinders the broad applicability of smoothing splines. In practice, the full-sample smoothing spline estimator can be approximated by an estimator based on $q$ randomly selected basis functions, resulting in a computational cost of $O(nq^2)$. It is known that these two estimators converge at the same rate when $q$ is of order $O\\{n^{2\/(pr+1)}\\}$, where $p\\in [1,2]$ depends on the true function and $r &amp;gt; 1$ depends on the type of spline. Such a $q$ is called the essential number of basis functions. In this article, we develop a more efficient basis selection method. By selecting basis functions corresponding to approximately equally spaced observations, the proposed method chooses a set of basis functions with great diversity. The asymptotic analysis shows that the proposed smoothing spline estimator can decrease $q$ to around $O\\{n^{1\/(pr+1)}\\}$ when $d\\leq pr+1$. Applications to synthetic and real-world datasets show that the proposed method leads to a smaller prediction error than other basis selection methods.<\/jats:p>","DOI":"10.1093\/biomet\/asaa019","type":"journal-article","created":{"date-parts":[[2020,3,6]],"date-time":"2020-03-06T20:20:40Z","timestamp":1583526040000},"page":"723-735","source":"Crossref","is-referenced-by-count":29,"title":["More efficient approximation of smoothing splines via space-filling basis selection"],"prefix":"10.1093","volume":"107","author":[{"given":"Cheng","family":"Meng","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Georgia, 310 Herty Dr., Athens, Georgia 30602, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinlian","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Georgia, 310 Herty Dr., Athens, Georgia 30602, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jingyi","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Georgia, 310 Herty Dr., Athens, Georgia 30602, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenxuan","family":"Zhong","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Georgia, 310 Herty Dr., Athens, Georgia 30602, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ping","family":"Ma","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Georgia, 310 Herty Dr., Athens, Georgia 30602, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2020,5,7]]},"reference":[{"key":"2020081714270044200_B1","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1080\/00031305.1992.10475879","article-title":"An introduction to kernel and nearest-neighbor nonparametric regression","volume":"46","author":"Altman,","year":"1992","journal-title":"Am. 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