{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T17:11:10Z","timestamp":1781629870254,"version":"3.54.5"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T00:00:00Z","timestamp":1768262400000},"content-version":"vor","delay-in-days":12,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100014597","name":"Universidade da Coru\u00f1a","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100014597","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Stat"],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Bandwidth selection is a central issue in kernel density estimation. For large datasets, classical selectors such as cross-validation and bootstrap become computationally intensive and may yield bandwidths with high variability. This paper proposes subagging-based versions of several popular selectors, including cross-validation, direct plug-in, and bootstrap methods. These selectors are constructed by computing bandwidths over multiple subsamples (without replacement), rescaling them, and averaging the results. We also introduce a novel regression-based approach, Regression Subbagging (RSB), which extrapolates the optimal bandwidth via a log-log regression, avoiding the need to assume a known convergence rate. We assess statistical accuracy in terms of the mean squared error of the selectors and the corresponding MISE of the resulting kernel estimators, using the optimal bandwidth as a benchmark. Computational efficiency is evaluated via parallel implementations using the  and  packages in , reporting speedups as a function of the number of CPU cores. The results confirm that subagging improves or preserves statistical performance while yielding substantial runtime reductions, especially for demanding selectors like cross-validation and bootstrap. The RSB variant, in particular, stands out as a scalable, flexible, and robust solution. The core methods are implemented in the  package , available on CRAN.<\/jats:p>","DOI":"10.1007\/s00180-025-01712-4","type":"journal-article","created":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T06:44:12Z","timestamp":1768286652000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Subagging for bandwidth selection: a computationally efficient approach to kernel density estimation"],"prefix":"10.1007","volume":"41","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9201-5423","authenticated-orcid":false,"given":"Mario","family":"Francisco-Fern\u00e1ndez","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4930-3603","authenticated-orcid":false,"given":"Daniel","family":"Barreiro-Ures","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8304-687X","authenticated-orcid":false,"given":"Ricardo","family":"Cao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,1,13]]},"reference":[{"key":"1712_CR1","unstructured":"Barreiro-Ures D (2021) Nonparametric density and regression estimation for samples of very large size. Phd thesis, Universidade da Coru\u00f1a. http:\/\/hdl.handle.net\/2183\/29370"},{"key":"1712_CR3","doi-asserted-by":"publisher","unstructured":"Barreiro-Ures D, Cao R, Francisco-Fern\u00e1ndez M (2023) Computational efficiency of bagging bootstrap bandwidth selection for density estimation with big data. In: Balakrishnan N, Gil M\u00c1, Mart\u00edn N, Morales D, Pardo MC (eds) Trends in Mathematical, Information and Data Sciences: A Tribute to Leandro Pardo. Studies in Systems, Decision and Control, vol. 445, pp 307\u2013317. Springer, Cham. https:\/\/doi.org\/10.1007\/978-3-031-04137-2_25","DOI":"10.1007\/978-3-031-04137-2_25"},{"issue":"4","key":"1712_CR2","doi-asserted-by":"publisher","first-page":"981","DOI":"10.1093\/biomet\/asaa092","volume":"108","author":"D Barreiro-Ures","year":"2021","unstructured":"Barreiro-Ures D, Cao R, Francisco-Fern\u00e1ndez M, Hart JD (2021) Bagging cross-validated bandwidths with application to big data. Biometrika 108(4):981\u2013988. https:\/\/doi.org\/10.1093\/biomet\/asaa092","journal-title":"Biometrika"},{"key":"1712_CR4","unstructured":"Barreiro-Ures D, Fernandez-Casal R, Hart J, Cao R, Francisco-Fernandez M (2024) Baggingbwsel: bagging bandwidth selection in kernel density and regression estimation. R package version 1.1. https:\/\/cran.r-project.org\/package=baggingbwsel"},{"key":"1712_CR5","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/BF00058655","volume":"24","author":"L Breiman","year":"1996","unstructured":"Breiman L (1996) Bagging predictors. Mach Learn 24:123\u2013140. https:\/\/doi.org\/10.1007\/BF00058655","journal-title":"Mach Learn"},{"issue":"4","key":"1712_CR6","doi-asserted-by":"publisher","first-page":"927","DOI":"10.1214\/aos\/1031689014","volume":"30","author":"P B\u00fchlmann","year":"2002","unstructured":"B\u00fchlmann P, Yu B (2002) Analyzing bagging. Ann Stat 30(4):927\u2013961. https:\/\/doi.org\/10.1214\/aos\/1031689014","journal-title":"Ann Stat"},{"issue":"1","key":"1712_CR7","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1006\/jmva.1993.1030","volume":"45","author":"R Cao","year":"1993","unstructured":"Cao R (1993) Bootstrapping the mean integrated squared error. J Multivar Anal 45(1):137\u2013160. https:\/\/doi.org\/10.1006\/jmva.1993.1030","journal-title":"J Multivar Anal"},{"issue":"3","key":"1712_CR8","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1016\/j.jspi.2006.06.002","volume":"137","author":"JH Friedman","year":"2007","unstructured":"Friedman JH, Hall P (2007) On bagging and nonlinear estimation. J Stat Plann Inference 137(3):669\u2013683. https:\/\/doi.org\/10.1016\/j.jspi.2006.06.002","journal-title":"J Stat Plann Inference"},{"key":"1712_CR9","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1007\/BF00363516","volume":"74","author":"P Hall","year":"1987","unstructured":"Hall P, Marron JS (1987) Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation. Probab Theory Relat Fields 74:567\u2013581. https:\/\/doi.org\/10.1007\/BF00363516","journal-title":"Probab Theory Relat Fields"},{"issue":"1","key":"1712_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01205233","volume":"92","author":"P Hall","year":"1992","unstructured":"Hall P, Marron JS, Park BU (1992) Smoothed cross-validation. Probab Theory Relat Fields 92(1):1\u201320. https:\/\/doi.org\/10.1007\/BF01205233","journal-title":"Probab Theory Relat Fields"},{"issue":"1","key":"1712_CR10","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1093\/biomet\/asn068","volume":"96","author":"P Hall","year":"2009","unstructured":"Hall P, Robinson AP (2009) Reducing variability of crossvalidation for smoothing parameter choice. Biometrika 96(1):175\u2013186. https:\/\/doi.org\/10.1093\/biomet\/asn068","journal-title":"Biometrika"},{"key":"1712_CR12","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1080\/07474938708800136","volume":"6","author":"JS Marron","year":"1987","unstructured":"Marron JS (1987) Partitioned cross-validation. Econ Rev 6:271\u2013283. https:\/\/doi.org\/10.1080\/07474938708800136","journal-title":"Econ Rev"},{"issue":"1","key":"1712_CR13","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1613\/jair.614","volume":"11","author":"DW Opitz","year":"1999","unstructured":"Opitz DW, Maclin R (1999) Popular ensemble methods: an empirical study. J Artif Intell Res 11(1):169\u2013198. https:\/\/doi.org\/10.1613\/jair.614","journal-title":"J Artif Intell Res"},{"issue":"3","key":"1712_CR14","doi-asserted-by":"publisher","first-page":"1065","DOI":"10.1214\/aoms\/1177704472","volume":"33","author":"E Parzen","year":"1962","unstructured":"Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33(3):1065\u20131076. https:\/\/doi.org\/10.1214\/aoms\/1177704472","journal-title":"Ann Math Stat"},{"issue":"3","key":"1712_CR15","doi-asserted-by":"publisher","first-page":"832","DOI":"10.1214\/aoms\/1177728190","volume":"27","author":"M Rosenblatt","year":"1956","unstructured":"Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Stat 27(3):832\u2013837. https:\/\/doi.org\/10.1214\/aoms\/1177728190","journal-title":"Ann Math Stat"},{"issue":"400","key":"1712_CR16","doi-asserted-by":"publisher","first-page":"1131","DOI":"10.1080\/01621459.1987.10478550","volume":"82","author":"DW Scott","year":"1987","unstructured":"Scott DW, Terrell GR (1987) Biased and unbiased cross-validation in density estimation. J Am Stat Assoc 82(400):1131\u20131146. https:\/\/doi.org\/10.1080\/01621459.1987.10478550","journal-title":"J Am Stat Assoc"},{"issue":"3","key":"1712_CR17","doi-asserted-by":"publisher","first-page":"683","DOI":"10.1111\/j.2517-6161.1991.tb01857.x","volume":"53","author":"S Sheather","year":"1991","unstructured":"Sheather S, Jones MC (1991) A reliable data-based bandwidth selection method for kernel density estimation. J Roy Stat Soc: Ser B (Methodol) 53(3):683\u2013690. https:\/\/doi.org\/10.1111\/j.2517-6161.1991.tb01857.x","journal-title":"J Roy Stat Soc: Ser B (Methodol)"},{"key":"1712_CR18","volume-title":"Density estimation for statistics and data analysis. Monographs on statistics and applied probability","author":"BW Silverman","year":"1986","unstructured":"Silverman BW (1986) Density estimation for statistics and data analysis. Monographs on statistics and applied probability. Chapman and Hall, London"},{"issue":"2","key":"1712_CR19","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1111\/j.2517-6161.1974.tb00994.x","volume":"36","author":"M Stone","year":"1974","unstructured":"Stone M (1974) Cross-validatory choice and assessment of statistical predictions. J Roy Stat Soc: Ser B (Methodol) 36(2):111\u2013147. https:\/\/doi.org\/10.1111\/j.2517-6161.1974.tb00994.x","journal-title":"J Roy Stat Soc: Ser B (Methodol)"},{"issue":"2","key":"1712_CR20","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1007\/s10463-023-00890-6","volume":"76","author":"MNM Van Lieshout","year":"2024","unstructured":"Van Lieshout MNM (2024) Non-parametric adaptive bandwidth selection for kernel estimators of spatial intensity functions. Ann Inst Stat Math 76(2):313\u2013331. https:\/\/doi.org\/10.1007\/s10463-023-00890-6","journal-title":"Ann Inst Stat Math"},{"key":"1712_CR21","doi-asserted-by":"crossref","unstructured":"Wand M (1995) KernSmooth: functions for kernel smoothing supporting Wand & Jones (2025). R package version 2.23-26. https:\/\/CRAN.R-project.org\/package=KernSmooth","DOI":"10.1007\/978-1-4899-4493-1"},{"key":"1712_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-4493-1","volume-title":"Kernel smoothing. Monographs on statistics and applied probability","author":"MP Wand","year":"1995","unstructured":"Wand MP, Jones MC (1995) Kernel smoothing. Monographs on statistics and applied probability. Chapman and Hall\/CRC, London"},{"key":"1712_CR23","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1016\/j.csda.2015.03.005","volume":"89","author":"Q Wang","year":"2015","unstructured":"Wang Q, Lindsay BG (2015) Improving cross-validated bandwidth selection using subsampling-extrapolation techniques. Comput Stat Data Anal 89:51\u201371. https:\/\/doi.org\/10.1016\/j.csda.2015.03.005","journal-title":"Comput Stat Data Anal"}],"container-title":["Computational Statistics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-025-01712-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00180-025-01712-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-025-01712-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T16:39:41Z","timestamp":1781627981000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00180-025-01712-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["1712"],"URL":"https:\/\/doi.org\/10.1007\/s00180-025-01712-4","relation":{},"ISSN":["0943-4062","1613-9658"],"issn-type":[{"value":"0943-4062","type":"print"},{"value":"1613-9658","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1]]},"assertion":[{"value":"16 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 January 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflict of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval and consent to participate"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}],"article-number":"29"}}