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Small sample inference for collections of Bernoulli trials

## Descriptive

TypeOfResource
Text
TitleInfo (ID = T-1)
Title
Small sample inference for collections of Bernoulli trials
SubTitle
PartName
PartNumber
NonSort
Identifier (displayLabel = ); (invalid = )
ETD_2319
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000052166
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eng
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theses
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Topic
Statistics and Biostatistics
Subject (ID = SBJ-2); (authority = ETD-LCSH)
Topic
Approximation theory
Subject (ID = SBJ-3); (authority = ETD-LCSH)
Topic
Bernoulli numbers
Abstract
This dissertation discusses two applied problems solved by saddlepoint approximation methods. The first part of this thesis concerns continuity corrected saddlepoint approximations for testing and confidence intervals for the difference of two independent binomial proportions. We propose two new continuity corrections, and compare them with the other four continuity corrections. Continuity corrections may give a more accurate approximations to the tail area of discrete random variables. To get a confidence interval for the difference of two independent binomial proportions, we proposed continuity corrections 1/2LCM(m,n) with the least common multiple (LCM) of the two binomial distributions sample sizes m and n considered by Xu and Kolassacite{Xu:2008}, and R/2LCM(m,n), an adjusted version of 1/2LCM(m,n) with R by Xu and Kolassacite{Xu:2009}, a heuristic factor calculator from the standard error of the marginal null binomial distribution. We compare exact coverage probabilities for intervals calculated by using saddlepoint approximation with different corrections. The primary criterion for evaluating the corrections is agreement of actual with nominal coverage probabilities. Because of the ordering of the continuity corrections, the coverage probabilities will be ordered similarly. For all the cases considered with minimum expected cell size of at least 1, numerical results indicate that R/2LCM(m,n) has coverage probabilities very close to the nominal 95% and 90% even for the minimum of sample sizes as small as 5-9, and R/2LCM(m,n) improved uncorrected saddlepoint approximation methods moderately for the nominal 95% and 90% intervals; however, that the Yates continuity correction (2m)^{-1}+(2n)^{-1} is unnecessarily conservative for 95% and 90% but reasonable for 99% intervals.
The second part of this thesis concerns the sequential likelihood ratio test using in computerized adaptive testing. We consider sequential testing techniques, including the truncated sequential probability ratio test and the Haybittle-Peto test. Both of these tests in their original forms rely on approximate normality of the signed roots of the log likelihood ratio tests, and approximate boundary crossing probabilities for discrete normal-theory random walks. Bartroff, Finkelman and Laicite{Bartroff:2008} modify these techniques by using Monte Carlo approximations to calibrate the truncation boundary. We propose a hybrid Monte Carlo-Asymptotic approach, in which we substitute an easy Monte Carlo approximation in place of boundary crossing probabilities for Brownian motions, and use asymptotic approximations for the distribution of the signed root of the likelihood ratio test statistic. We found that after selecting stopping boundaries using normal-based Monte Carlo calculations, reliance on asymptotic normality
of the signed root of the log likelihood ratio statistics provided adequate control of Type I error, without recourse to more complicated Monte Carlo operations. We also observe markable improvement using
Barndorff-Nielsen's r* formula (Barndorff-Nielsen, 1991).
PhysicalDescription
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electronic resource
Extent
xi, 58 p. : ill.
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application/pdf
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text/xml
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references (p. 55-57)
Note (type = statement of responsibility)
by Lu Xu
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Xu
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Lu
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1979-
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Lu Xu
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Kolassa
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John
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chair
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John E. Kolassa
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Xie
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Minge
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internal member
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Minge Xie
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Hoover
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Donald
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Donald R. Hoover
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Ohman-Strickland
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Pamela
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outside member
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Pamela Ohman-Strickland
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Rutgers University
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degree grantor
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2010
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2010-01
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xx
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Title
Rutgers University Electronic Theses and Dissertations
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ETD
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Title
Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3NK3F5M
Genre (authority = ExL-Esploro)
ETD doctoral
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## Rights

RightsDeclaration (AUTHORITY = GS); (ID = rulibRdec0006)
The author owns the copyright to this work.
Status
Notice
Note
Availability
Status
Open
Reason
Note
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Name
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Xu
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Lu
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2009-12-20 11:03:56
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Lu Xu
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Rutgers University. Graduate School - New Brunswick
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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