{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/23054"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/23054","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Sequential confidence bands for densities","abstract":"We propose a fully sequential procedure for constructing a fixed width confidence band for an unknown density on a finite interval and show the procedure has the desired coverage probability asymptotically as the width of the band approaches zero. The procedure is based on a result of Bickel and Rosenblatt (1973, Ann. Statist. 1, 1071-1095). Its implementation in the sequential setting cannot be obtained using Anscombe's theorem, because the normalized maximal deviations between the kernel estimate and the true density are not uniformly continuous in probability (u.c.i.p.). Instead, we obtain a slightly weaker version of the u.c.i.p. property and a correspondingly stronger convergence property of the stopping rule. These together yield the desired results. We also present some simulation results and applications of the basic method to stopping rules of interest. Similar result is also obtained for censored data case.","abstract_html":"We propose a fully sequential procedure for constructing a fixed width confidence band for an unknown density on a finite interval and show the procedure has the desired coverage probability asymptotically as the width of the band approaches zero. The procedure is based on a result of Bickel and Rosenblatt (1973, Ann. Statist. 1, 1071-1095). Its implementation in the sequential setting cannot be obtained using Anscombe's theorem, because the normalized maximal deviations between the kernel estimate and the true density are not uniformly continuous in probability (u.c.i.p.). Instead, we obtain a slightly weaker version of the u.c.i.p. property and a correspondingly stronger convergence property of the stopping rule. These together yield the desired results. We also present some simulation results and applications of the basic method to stopping rules of interest. Similar result is also obtained for censored data case.","abstract_has_math":false,"creators":["Xu, Yi"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Statistics","degree_department":null,"school":null,"contributors":["Martinsek, Adam T."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T14:00:30Z","date_published":"2011-05-07T14:00:30Z","updated_at":"2026-07-22T22:25:21Z","subjects":["Statistics"],"languages":["eng"],"rights":["Copyright 1995 Xu, Yi"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9543778","(UMI)AAI9543778"],"render_values":[{"text":"AAI9543778","href":null,"code":true},{"text":"(UMI)AAI9543778","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/23054","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Martinsek, Adam T."]},{"key":"dc:creator","label":"Author","values":["Xu, Yi"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T14:00:30Z","10000-01-01","1995"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Statistics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1995 Xu, Yi"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9543778","(UMI)AAI9543778","http://hdl.handle.net/2142/23054"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We propose a fully sequential procedure for constructing a fixed width confidence band for an unknown density on a finite interval and show the procedure has the desired coverage probability asymptotically as the width of the band approaches zero. The procedure is based on a result of Bickel and Rosenblatt (1973, Ann. Statist. 1, 1071-1095). Its implementation in the sequential setting cannot be obtained using Anscombe's theorem, because the normalized maximal deviations between the kernel estimate and the true density are not uniformly continuous in probability (u.c.i.p.). Instead, we obtain a slightly weaker version of the u.c.i.p. property and a correspondingly stronger convergence property of the stopping rule. These together yield the desired results. We also present some simulation results and applications of the basic method to stopping rules of interest. Similar result is also obtained for censored data case.","Made available in DSpace on 2011-05-07T14:00:30Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9543778.pdf: 1277107 bytes, checksum: b512fee14f427c9bc007dc8b01bb6010 (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:01:52Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:29:22-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Sequential confidence bands for densities"]}]}],"canonical_facts":{"dc:contributor":["Martinsek, Adam T."],"dc:creator":["Xu, Yi"],"dc:date":["2011-05-07T14:00:30Z","10000-01-01","1995"],"dc:description":["We propose a fully sequential procedure for constructing a fixed width confidence band for an unknown density on a finite interval and show the procedure has the desired coverage probability asymptotically as the width of the band approaches zero. The procedure is based on a result of Bickel and Rosenblatt (1973, Ann. Statist. 1, 1071-1095). Its implementation in the sequential setting cannot be obtained using Anscombe's theorem, because the normalized maximal deviations between the kernel estimate and the true density are not uniformly continuous in probability (u.c.i.p.). Instead, we obtain a slightly weaker version of the u.c.i.p. property and a correspondingly stronger convergence property of the stopping rule. These together yield the desired results. We also present some simulation results and applications of the basic method to stopping rules of interest. Similar result is also obtained for censored data case.","Made available in DSpace on 2011-05-07T14:00:30Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9543778.pdf: 1277107 bytes, checksum: b512fee14f427c9bc007dc8b01bb6010 (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:01:52Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:29:22-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9543778","(UMI)AAI9543778","http://hdl.handle.net/2142/23054"],"dc:language":["eng"],"dc:rights":["Copyright 1995 Xu, Yi"],"dc:subject":["Statistics"],"dc:title":["Sequential confidence bands for densities"],"dc:type":["text"],"thesis:degree_discipline":["Statistics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:21Z"}