{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,12]],"date-time":"2026-04-12T06:50:51Z","timestamp":1775976651939,"version":"3.50.1"},"reference-count":14,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T00:00:00Z","timestamp":1625529600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Simultaneous confidence intervals are commonly used in joint inference of multiple parameters. When the underlying joint distribution of the estimates is unknown, nonparametric methods can be applied to provide distribution-free simultaneous confidence intervals. In this note, we propose new one-sided and two-sided nonparametric simultaneous confidence intervals based on the percentile bootstrap approach. The admissibility of the proposed intervals is established. The numerical results demonstrate that the proposed confidence intervals maintain the correct coverage probability for both normal and non-normal distributions. For smoothed bootstrap estimates, we extend Efron\u2019s (2014) nonparametric delta method to construct nonparametric simultaneous confidence intervals. The methods are applied to construct simultaneous confidence intervals for LASSO regression estimates.<\/jats:p>","DOI":"10.3390\/sym13071212","type":"journal-article","created":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T23:53:51Z","timestamp":1625615631000},"page":"1212","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On the Admissibility of Simultaneous Bootstrap Confidence Intervals"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8057-5100","authenticated-orcid":false,"given":"Xin","family":"Gao","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5674-2076","authenticated-orcid":false,"given":"Frank","family":"Konietschke","sequence":"additional","affiliation":[{"name":"Berlin Institute of Health, Institute of Biometry and Clinical Epidemiology Berlin and Charit\u00e9-Universit\u00e4tsmedizin Berlin, Corporate Member of Freie Universit\u00e4t Berlin, Humboldt-Universit\u00e4t zu Berlin, 10117 Berlin, Germany"},{"name":"Berlin Institute of Health (BIH), Anna-Louisa-Karsch-Stra\u00dfe 2, 10178 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiong","family":"Li","sequence":"additional","affiliation":[{"name":"BNU-HKBU United International College, Zhuhai 519087, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"399","DOI":"10.2307\/2348711","article-title":"Confidence intervals versus regions","volume":"44","author":"Draper","year":"1995","journal-title":"Statistician"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/aos\/1176344552","article-title":"Bootstrap methods: Another look at the Jackknife","volume":"7","author":"Efron","year":"1979","journal-title":"Ann. 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B Methodol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1080\/01621459.2013.823775","article-title":"Estimation and Accuracy after Model Selection","volume":"109","author":"Efron","year":"2014","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2431","DOI":"10.1214\/aos\/1032181161","article-title":"Using Specially Designed Exponential Families for Density Estimation","volume":"24","author":"Efron","year":"1996","journal-title":"Ann. Stat."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/BF00058655","article-title":"Bagging Predictors","volume":"24","author":"Breiman","year":"1996","journal-title":"Mach. Learn."},{"key":"ref_10","first-page":"323","article-title":"Observations on Bagging","volume":"16","author":"Buja","year":"2006","journal-title":"Stat. Sin."},{"key":"ref_11","unstructured":"Jaeckel, L. (1972). The Infinitesimal Jackknife, Bell Laboratory."},{"key":"ref_12","unstructured":"Hothorn, T., Bretz, F., Westfall, P., Heiberger, R.M., Schuetzenmeister, A., Scheibe, S., and Hothorn, M.T. (2016). Package \u2018multcomp\u2019: Simultaneous Inference in General Parametric Models. Project for Statistical Computing, CRC Press."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hastie, T., Tibshirani, R., and Friedman, J. (2001). Elements of Statistical Learning; Data Mining, Inference and Prediction, Springer.","DOI":"10.1007\/978-0-387-21606-5"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1198\/016214504000001907","article-title":"False discovery rate\u2013adjusted multiple confidence intervals for selected parameters","volume":"100","author":"Benjamini","year":"2005","journal-title":"J. Am. Stat. 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