{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/21469"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/21469","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Some sequential estimation problems in logistic regression models","abstract":"Let $({\\bf X}\\sb{i},Y\\sb{i}), i = 1,2,\\cdots,$ be a random sample satisfying a logistic regression model; that is, for each i, log($P(Y\\sb{i}$ = $1\\vert{\\bf X}\\sb{i})/P(Y\\sb{i}$ = 0$\\vert{\\bf X}\\sb{i})\\rbrack$ = ${\\bf X}\\sbsp{i}{T}\\beta\\sb0,$ where $Y\\sb{i}\\in\\{$0,1$\\},$ ${\\bf X}\\sb{i}\\in{\\bf R}\\sp{p}$ and $\\beta\\sb0\\in{\\bf R}\\sp{p}$ is the unknown parameter vector of the logistic regression model. It is known that $\\sqrt{n}(\\\\beta\\sb n-\\beta\\sb0){\\buildrel{\\cal L}\\over{\\longrightarrow}} N(0\\sb p,\\Sigma\\sp{-1}),$ where $\\\\beta\\sb n$ is a MLE of $\\beta\\sb0$ and $\\Sigma\\sp{-1}$ is the Fisher information matrix. If $\\Sigma$ is known then $R\\sb d=\\{Z\\in{\\bf R}\\sp p:n(Z-\\\\beta\\sb n)\\sp T\\Sigma(Z-\\\\beta\\sb n)$ $\\le n\\lambda d\\sp2\\}$ defines a confidence ellipsoid for $\\beta\\sb0$, with maximum axis $\\le 2d$ and $P(\\beta\\sb0\\in R\\sb d)\\approx 1 - \\alpha$ provided $n\\ge a\\sp2/(\\lambda d\\sp2),$ where $\\lambda$ is the smallest eigenvalue of $\\Sigma$ and a satisfies $P(\\chi\\sp2(p)\\le a\\sp2)$ = $1 - \\alpha$. If $\\Sigma$ is unknown then $\\lambda$ usually will be unknown. Hence, there is no fixed sample size that can be used to construct a confidence ellipsoid with prescribed accuracy and confidence level. In this work, a sequential procedure is proposed to overcome this difficulty. The procedure is shown to be asymptotically consistent and efficient. That is to say, as d approaches 0 the coverage probability converges to the required confidence level and the ratio of the expected sample size to the unknown best fixed sample size converges to 1. Similar asymptotic properties for fixed proportional accuracy problems and for two stage procedures have also been obtained.","abstract_html":"Let $({\\bf X}\\sb{i},Y\\sb{i}), i = 1,2,\\cdots,$ be a random sample satisfying a logistic regression model; that is, for each i, log($P(Y\\sb{i}$ = $1\\vert{\\bf X}\\sb{i})/P(Y\\sb{i}$ = 0$\\vert{\\bf X}\\sb{i})\\rbrack$ = <span class=\"etd-inline-math\">{\\bf X}\\sbsp{i}{T}&beta;\\sb0,</span> where $Y\\sb{i}\\in\\{$0,1$\\},$ ${\\bf X}\\sb{i}\\in{\\bf R}\\sp{p}$ and <span class=\"etd-inline-math\">&beta;\\sb0\\in{\\bf R}\\sp{p}</span> is the unknown parameter vector of the logistic regression model. It is known that <span class=\"etd-inline-math\">\\sqrt{n}(\\&beta;\\sb n-&beta;\\sb0){\\buildrel{\\cal L}\\over{\\longrightarrow}} N(0\\sb p,\\Sigma\\sp{-1}),</span> where <span class=\"etd-inline-math\">\\&beta;\\sb n</span> is a MLE of <span class=\"etd-inline-math\">&beta;\\sb0</span> and $\\Sigma\\sp{-1}$ is the Fisher information matrix. If $\\Sigma$ is known then <span class=\"etd-inline-math\">R\\sb d=\\{Z\\in{\\bf R}\\sp p:n(Z-\\&beta;\\sb n)\\sp T\\Sigma(Z-\\&beta;\\sb n)</span> $\\le n\\lambda d\\sp2\\}$ defines a confidence ellipsoid for <span class=\"etd-inline-math\">&beta;\\sb0</span>, with maximum axis $\\le 2d$ and <span class=\"etd-inline-math\">P(&beta;\\sb0\\in R\\sb d)\\approx 1 - &alpha;</span> provided $n\\ge a\\sp2/(\\lambda d\\sp2),$ where $\\lambda$ is the smallest eigenvalue of $\\Sigma$ and a satisfies $P(\\chi\\sp2(p)\\le a\\sp2)$ = <span class=\"etd-inline-math\">1 - &alpha;</span>. If $\\Sigma$ is unknown then $\\lambda$ usually will be unknown. Hence, there is no fixed sample size that can be used to construct a confidence ellipsoid with prescribed accuracy and confidence level. In this work, a sequential procedure is proposed to overcome this difficulty. The procedure is shown to be asymptotically consistent and efficient. That is to say, as d approaches 0 the coverage probability converges to the required confidence level and the ratio of the expected sample size to the unknown best fixed sample size converges to 1. Similar asymptotic properties for fixed proportional accuracy problems and for two stage procedures have also been obtained.","abstract_has_math":true,"creators":["Chang, Yuan-Chin Ivan"],"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":"2011-05-07T13:09:32Z","date_published":"2011-05-07T13:09:32Z","updated_at":"2026-07-22T22:25:18Z","subjects":["Statistics"],"languages":["eng"],"rights":["Copyright 1991 Chang, Yuan-Chin Ivan"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9136567","(UMI)AAI9136567"],"render_values":[{"text":"AAI9136567","href":null,"code":true},{"text":"(UMI)AAI9136567","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/21469","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":["Chang, Yuan-Chin Ivan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:09:32Z","10000-01-01","1991"]},{"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 1991 Chang, Yuan-Chin Ivan"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9136567","(UMI)AAI9136567","http://hdl.handle.net/2142/21469"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let $({\\bf X}\\sb{i},Y\\sb{i}), i = 1,2,\\cdots,$ be a random sample satisfying a logistic regression model; that is, for each i, log($P(Y\\sb{i}$ = $1\\vert{\\bf X}\\sb{i})/P(Y\\sb{i}$ = 0$\\vert{\\bf X}\\sb{i})\\rbrack$ = ${\\bf X}\\sbsp{i}{T}\\beta\\sb0,$ where $Y\\sb{i}\\in\\{$0,1$\\},$ ${\\bf X}\\sb{i}\\in{\\bf R}\\sp{p}$ and $\\beta\\sb0\\in{\\bf R}\\sp{p}$ is the unknown parameter vector of the logistic regression model. It is known that $\\sqrt{n}(\\\\beta\\sb n-\\beta\\sb0){\\buildrel{\\cal L}\\over{\\longrightarrow}} N(0\\sb p,\\Sigma\\sp{-1}),$ where $\\\\beta\\sb n$ is a MLE of $\\beta\\sb0$ and $\\Sigma\\sp{-1}$ is the Fisher information matrix. If $\\Sigma$ is known then $R\\sb d=\\{Z\\in{\\bf R}\\sp p:n(Z-\\\\beta\\sb n)\\sp T\\Sigma(Z-\\\\beta\\sb n)$ $\\le n\\lambda d\\sp2\\}$ defines a confidence ellipsoid for $\\beta\\sb0$, with maximum axis $\\le 2d$ and $P(\\beta\\sb0\\in R\\sb d)\\approx 1 - \\alpha$ provided $n\\ge a\\sp2/(\\lambda d\\sp2),$ where $\\lambda$ is the smallest eigenvalue of $\\Sigma$ and a satisfies $P(\\chi\\sp2(p)\\le a\\sp2)$ = $1 - \\alpha$. If $\\Sigma$ is unknown then $\\lambda$ usually will be unknown. Hence, there is no fixed sample size that can be used to construct a confidence ellipsoid with prescribed accuracy and confidence level. In this work, a sequential procedure is proposed to overcome this difficulty. The procedure is shown to be asymptotically consistent and efficient. That is to say, as d approaches 0 the coverage probability converges to the required confidence level and the ratio of the expected sample size to the unknown best fixed sample size converges to 1. Similar asymptotic properties for fixed proportional accuracy problems and for two stage procedures have also been obtained.","Made available in DSpace on 2011-05-07T13:09:32Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9136567.pdf: 2036907 bytes, checksum: 2a79fa5d556b9611a9e84cee20f5afb3 (MD5) Previous issue date: 1991","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:50:59Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:23:22-05:00 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":["Some sequential estimation problems in logistic regression models"]}]}],"canonical_facts":{"dc:contributor":["Martinsek, Adam T."],"dc:creator":["Chang, Yuan-Chin Ivan"],"dc:date":["2011-05-07T13:09:32Z","10000-01-01","1991"],"dc:description":["Let $({\\bf X}\\sb{i},Y\\sb{i}), i = 1,2,\\cdots,$ be a random sample satisfying a logistic regression model; that is, for each i, log($P(Y\\sb{i}$ = $1\\vert{\\bf X}\\sb{i})/P(Y\\sb{i}$ = 0$\\vert{\\bf X}\\sb{i})\\rbrack$ = ${\\bf X}\\sbsp{i}{T}\\beta\\sb0,$ where $Y\\sb{i}\\in\\{$0,1$\\},$ ${\\bf X}\\sb{i}\\in{\\bf R}\\sp{p}$ and $\\beta\\sb0\\in{\\bf R}\\sp{p}$ is the unknown parameter vector of the logistic regression model. It is known that $\\sqrt{n}(\\\\beta\\sb n-\\beta\\sb0){\\buildrel{\\cal L}\\over{\\longrightarrow}} N(0\\sb p,\\Sigma\\sp{-1}),$ where $\\\\beta\\sb n$ is a MLE of $\\beta\\sb0$ and $\\Sigma\\sp{-1}$ is the Fisher information matrix. If $\\Sigma$ is known then $R\\sb d=\\{Z\\in{\\bf R}\\sp p:n(Z-\\\\beta\\sb n)\\sp T\\Sigma(Z-\\\\beta\\sb n)$ $\\le n\\lambda d\\sp2\\}$ defines a confidence ellipsoid for $\\beta\\sb0$, with maximum axis $\\le 2d$ and $P(\\beta\\sb0\\in R\\sb d)\\approx 1 - \\alpha$ provided $n\\ge a\\sp2/(\\lambda d\\sp2),$ where $\\lambda$ is the smallest eigenvalue of $\\Sigma$ and a satisfies $P(\\chi\\sp2(p)\\le a\\sp2)$ = $1 - \\alpha$. If $\\Sigma$ is unknown then $\\lambda$ usually will be unknown. Hence, there is no fixed sample size that can be used to construct a confidence ellipsoid with prescribed accuracy and confidence level. In this work, a sequential procedure is proposed to overcome this difficulty. The procedure is shown to be asymptotically consistent and efficient. That is to say, as d approaches 0 the coverage probability converges to the required confidence level and the ratio of the expected sample size to the unknown best fixed sample size converges to 1. Similar asymptotic properties for fixed proportional accuracy problems and for two stage procedures have also been obtained.","Made available in DSpace on 2011-05-07T13:09:32Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9136567.pdf: 2036907 bytes, checksum: 2a79fa5d556b9611a9e84cee20f5afb3 (MD5) Previous issue date: 1991","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:50:59Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:23:22-05:00 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":["AAI9136567","(UMI)AAI9136567","http://hdl.handle.net/2142/21469"],"dc:language":["eng"],"dc:rights":["Copyright 1991 Chang, Yuan-Chin Ivan"],"dc:subject":["Statistics"],"dc:title":["Some sequential estimation problems in logistic regression models"],"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":"2026-07-22T22:25:18Z"}