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Chang, Yuan-Chin Ivan 2011-05-07T13:09:32Z 2011-05-07T13:09:32Z 10000-01-01 1991 Martinsek, Adam T. 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 AAI9136567 (UMI)AAI9136567 http://hdl.handle.net/2142/21469 eng Copyright 1991 Chang, Yuan-Chin Ivan Statistics Some sequential estimation problems in logistic regression models text Statistics Statistics University of Illinois at Urbana-Champaign Dissertation Ph.D.