{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T18:55:21Z","timestamp":1784141721641,"version":"3.55.0"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"359","license":[{"start":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T00:00:00Z","timestamp":1776297600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["10.55776\/P34808"],"award-info":[{"award-number":["10.55776\/P34808"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We study the sample median of independently generated quasi-Monte Carlo estimators based on randomized digital nets and prove it approximates the target integral value at almost the optimal convergence rate for various function spaces. In contrast to previous methods, the algorithm does not require a priori knowledge of underlying function spaces or even an input of pre-designed\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis t comma m comma s right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(t,m,s)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -digital nets, and is therefore easier to implement. This study provides further evidence that some types of randomized quasi-Monte Carlo estimators are heavy-tailed when applied to smooth integrands and taking the median can significantly improve the error by filtering out the outliers.\n                  <\/p>","DOI":"10.1090\/mcom\/4093","type":"journal-article","created":{"date-parts":[[2025,3,26]],"date-time":"2025-03-26T14:19:56Z","timestamp":1742998796000},"page":"1415-1446","source":"Crossref","is-referenced-by-count":7,"title":["Automatic optimal-rate convergence of randomized nets using median-of-means"],"prefix":"10.1090","volume":"95","author":[{"given":"Zexin","family":"Pan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,4,16]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"H. Bahouri, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, 2011.","DOI":"10.1007\/978-3-642-16830-7"},{"key":"2","doi-asserted-by":"crossref","unstructured":"R. E. Caflisch, W. Morokoff, and A. B. Owen, Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, J. Computat. Finance 1 (1997), 27\u201346.","DOI":"10.21314\/JCF.1997.005"},{"key":"3","doi-asserted-by":"crossref","unstructured":"L. Chen, M. Xu, and H. Zhang, A random integration algorithm for high-dimensional function spaces, Preprint (2025).","DOI":"10.1090\/mcom\/4171"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1007\/s10231-007-0048-z","article-title":"Koksma-Hlawka type inequalities of fractional order","volume":"187","author":"Dick, Josef","year":"2008","journal-title":"Ann. Mat. Pura Appl. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-3114","issn-type":"print"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"1519","DOI":"10.1137\/060666639","article-title":"Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order","volume":"46","author":"Dick, Josef","year":"2008","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"1372","DOI":"10.1214\/11-AOS880","article-title":"Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands","volume":"39","author":"Dick, Josef","year":"2011","journal-title":"Ann. 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M\u00e9tivier, Mean estimation for randomized quasi monte carlo method, Hal preprint hal-03631879v2 (2022)."},{"key":"11","unstructured":"T. Goda and D. Krieg, A simple universal algorithm for high-dimensional integration, Preprint (2024)."},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1137\/22M1525077","article-title":"A universal median quasi\u2013Monte Carlo integration","volume":"62","author":"Goda, Takashi","year":"2024","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"13","isbn-type":"print","first-page":"50","article-title":"Optimal summation and integration by deterministic, randomized, and quantum algorithms","author":"Heinrich, Stefan","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/354042718X"},{"key":"14","doi-asserted-by":"crossref","unstructured":"F. J. 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Tuffin, Confidence intervals for randomized quasi-Monte Carlo estimators, Tech. report, hal-04088085, 2023.","DOI":"10.1109\/WSC60868.2023.10408613"},{"issue":"474","key":"17","doi-asserted-by":"publisher","first-page":"712","DOI":"10.1198\/016214505000001410","article-title":"Estimating mean dimensionality of analysis of variance decompositions","volume":"101","author":"Liu, Ruixue","year":"2006","journal-title":"J. Amer. Statist. Assoc.","ISSN":"https:\/\/id.crossref.org\/issn\/0162-1459","issn-type":"print"},{"key":"18","doi-asserted-by":"crossref","unstructured":"Y. Liu, Randomized quasi-monte carlo and owen\u2019s boundary growth condition: A spectral analysis, Preprint (2025).","DOI":"10.1093\/imanum\/draf020"},{"issue":"4","key":"19","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1006\/jcom.1998.0489","article-title":"On the \ud835\udc3f\u2082-discrepancy for anchored boxes","volume":"14","author":"Matou\u0161ek, Ji\u0159\u00ed","year":"1998","journal-title":"J. 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Wang, Median QMC method for unbounded integrands over \u211d^{\ud835\udd64} in unanchored weighted sobolev spaces, Preprint (2025)."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-359\/S0025-5718-2025-04093-6\/mcom4093_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-359\/S0025-5718-2025-04093-6\/S0025-5718-2025-04093-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:02:05Z","timestamp":1776837725000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-359\/S0025-5718-2025-04093-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,16]]},"references-count":30,"journal-issue":{"issue":"359","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["S0025-5718-2025-04093-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4093","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2025,4,16]]}}}