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Higher order conditional inference using parallels with approximate Bayesian techniques
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Title
Higher order conditional inference using parallels with approximate Bayesian techniques
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ETD_1041
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000050480
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eng
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theses
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Statistics and Biostatistics
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Mathematical statistics
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Probabilities
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(authority = ETD-LCSH)
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Bayesian statistical decision theory
Abstract
I consider parametric models with a scalar parameter of
interest and multiple nuisance parameters. The likelihood ratio statistic is frequently used in statistical inference. The standard normal approximation to the likelihood ratio statistic generally has error of order O(n-1/2), where "n" denotes the sample size. When "n" is small, the normal approximation may not be adequate to do accurate inference. In practice, the true error is more important than asymptotic order. The intention of this study is to find an approximation which is relatively easy to apply, but which is accurate under small sample size settings. Saddlepoint approximations are well-known for higher order accuracy properties and remarkably good relative error properties. There are several saddlepoint approximations. I look for one that is flexible in
application while keeping a satisfactory convergence rate.
I evaluate, via Monte Carlo, the accuracies of several saddlepoint approximations, and of some classical methods, when these approximations are used to approximate p-values for hypotheses about a scalar parameter. Based on the results, I find that DiCiccio and Martin's (1993) approximations are interesting and deserve more research. Approximations of DiCiccio and Martin (1993) involve exploiting the parallels between Bayesian and frequentist inference, and can be constructed from general log-likelihood functions with relatively easy calculation, while keep the accuracy property.
Two difficulties arise in the application of these approximations. One is the instability around a singularity. The other and far more significant is the construction of the prior density functions utilized in these approximations. These prior density functions are
also called matching priors.
To make DiCiccio and Martin's (1993) approximations applicable in practice, I successfully resolve the above two problems. I remove the instability and fix the numerical difficulties in applying these approximations. The matching prior is the solution to a first order partial differential equation. The solution of this equation is
non-trivial under the general parametrization. I use a procedure to solve the equation numerically given any initial conditions.
As a conclusion, I suggest the use of DiCiccio and Martin's (1993) approximations with the construction procedure and the correction that I provide in this thesis.
interest and multiple nuisance parameters. The likelihood ratio statistic is frequently used in statistical inference. The standard normal approximation to the likelihood ratio statistic generally has error of order O(n-1/2), where "n" denotes the sample size. When "n" is small, the normal approximation may not be adequate to do accurate inference. In practice, the true error is more important than asymptotic order. The intention of this study is to find an approximation which is relatively easy to apply, but which is accurate under small sample size settings. Saddlepoint approximations are well-known for higher order accuracy properties and remarkably good relative error properties. There are several saddlepoint approximations. I look for one that is flexible in
application while keeping a satisfactory convergence rate.
I evaluate, via Monte Carlo, the accuracies of several saddlepoint approximations, and of some classical methods, when these approximations are used to approximate p-values for hypotheses about a scalar parameter. Based on the results, I find that DiCiccio and Martin's (1993) approximations are interesting and deserve more research. Approximations of DiCiccio and Martin (1993) involve exploiting the parallels between Bayesian and frequentist inference, and can be constructed from general log-likelihood functions with relatively easy calculation, while keep the accuracy property.
Two difficulties arise in the application of these approximations. One is the instability around a singularity. The other and far more significant is the construction of the prior density functions utilized in these approximations. These prior density functions are
also called matching priors.
To make DiCiccio and Martin's (1993) approximations applicable in practice, I successfully resolve the above two problems. I remove the instability and fix the numerical difficulties in applying these approximations. The matching prior is the solution to a first order partial differential equation. The solution of this equation is
non-trivial under the general parametrization. I use a procedure to solve the equation numerically given any initial conditions.
As a conclusion, I suggest the use of DiCiccio and Martin's (1993) approximations with the construction procedure and the correction that I provide in this thesis.
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xi, 62 p. : ill.
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Ph.D.
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Includes bibliographical references (p. 53-55)
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by Juan Zhang
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Zhang
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Juan
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Juan Zhang
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Kolassa
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John
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John E. Kolassa
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Cohen
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Arthur
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Arthur Cohen
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William
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William E. Strawderman
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Scheetz
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Linda
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outside member
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Linda J. Scheetz
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Rutgers University
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Graduate School - New Brunswick
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2008
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2008-10
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xx
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Rutgers University Electronic Theses and Dissertations
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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doi:10.7282/T3Q52PXG
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ETD doctoral
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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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