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Some properties of robust statistics under asymmetric models
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Text
TitleInfo
Title
Some properties of robust statistics under asymmetric models
Identifier
ETD_1121
Identifier
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000050467
Language
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eng
Genre
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theses
Subject
(authority = RUETD)
Topic
Statistics and Biostatistics
Subject
(authority = ETD-LCSH)
Topic
Robust statistics
Abstract
Properties of robust statistics have been extensively studied in the univariate setting when the underlying model is presumed to be symmetric, and in the multivariate case when the underlying model is presumed to be elliptically symmetric. Much less attention has been given to the behavior of robust statistics under asymmetric models. The goal of this dissertation is thus to obtain theoretical results for robust statistics under asymmetric models. To this end, local asymmetric alternatives to symmetric and elliptically symmetric distributions are considered. A key tool used in obtaining the theories presented in this dissertation is the LeCam's lemmas on contiguity.
The classes of robust univariate statistic considered here are the M-estimates, one-step version of the M-estimates, the W-estimates and the trimmed means. The classes of robust multivariate statistics considered are the M-estimates, the S-estimates, the CM-estimates and the MM-estimates, which are all treated under the unified framework of M-estimates with auxiliary scale, as well as their one-step versions. Asymptotic distributions of these statistics are obtained under local mixture models and skew-symmetric models. The asymptotic properties for the MM-estimates, even under elliptical symmetry, are the first such results for the multivariate MM-estimates.
Under asymmetry, different robust statistics for location are not consistent with each other, i.e. they are estimating different notions of central tendency. Likewise, in the multivariate setting, under non-elliptical distributions, the different scatter statistics are again not consistent with each other and are reflecting different structures of the underlying distribution. This suggests the difference in location statistics can be used to detect asymmetry and the comparison of different scatter statistics can be used to detect deviations from elliptical symmetry.
Consequently, new classes of tests for symmetry and for elliptical symmetry are introduced in this dissertation based upon the comparisons of different location statistics and different scatter statistics respectively. Furthermore, the asymptotic null distributions of the proposed test statistics are derived as well as their local power functions under contiguous mixture distributions. The local power functions help provide some guidelines for choosing the proper tuning constant of the proposed tests.
The classes of robust univariate statistic considered here are the M-estimates, one-step version of the M-estimates, the W-estimates and the trimmed means. The classes of robust multivariate statistics considered are the M-estimates, the S-estimates, the CM-estimates and the MM-estimates, which are all treated under the unified framework of M-estimates with auxiliary scale, as well as their one-step versions. Asymptotic distributions of these statistics are obtained under local mixture models and skew-symmetric models. The asymptotic properties for the MM-estimates, even under elliptical symmetry, are the first such results for the multivariate MM-estimates.
Under asymmetry, different robust statistics for location are not consistent with each other, i.e. they are estimating different notions of central tendency. Likewise, in the multivariate setting, under non-elliptical distributions, the different scatter statistics are again not consistent with each other and are reflecting different structures of the underlying distribution. This suggests the difference in location statistics can be used to detect asymmetry and the comparison of different scatter statistics can be used to detect deviations from elliptical symmetry.
Consequently, new classes of tests for symmetry and for elliptical symmetry are introduced in this dissertation based upon the comparisons of different location statistics and different scatter statistics respectively. Furthermore, the asymptotic null distributions of the proposed test statistics are derived as well as their local power functions under contiguous mixture distributions. The local power functions help provide some guidelines for choosing the proper tuning constant of the proposed tests.
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x, 96 p. : ill.
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Note
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Ph.D.
Note
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Includes bibliographical references (p. 93-95)
Note
(type = statement of responsibility)
by Jue Wang
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Wang
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Jue (Jue A.)
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author
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Jue Wang
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Tyler
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David
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chair
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Advisory Committee
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David E. Tyler
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Kolassa
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John
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internal member
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Advisory Committee
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John E. Kolassa
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Cabrera
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Javier
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internal member
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Advisory Committee
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Javier Cabrera
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Fernholz
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Luisa
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outside member
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Advisory Committee
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Luisa T. Fernholz
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Rutgers University
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degree grantor
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Graduate School - New Brunswick
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school
OriginInfo
DateCreated
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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ETD
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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Identifier
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doi:10.7282/T3C24WQW
Genre
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ETD doctoral
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Copyright protected
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Open
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Non-exclusive ETD license
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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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