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New results in probability bounding, a convexity statement and unimodality of multivariate discrete distributions

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TitleInfo (ID = T-1)
Title
New results in probability bounding, a convexity statement and unimodality of multivariate discrete distributions
Identifier (type = hdl)
http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.17395
Identifier
ETD_856
Language
LanguageTerm
English
Genre (authority = marcgt)
theses
Subject (ID = SBJ-1); (authority = RUETD)
Topic
Operations Research
Subject (ID = SBJ-2); (authority = ETD-LCSH)
Topic
Distribution (Probability theory)
Subject (ID = SBJ-3); (authority = ETD-LCSH)
Topic
Multivariate analysis
Abstract
This report constitutes the Doctoral Dissertation for Munevver Mine Subasi and consists of three topics: sharp bounds for the probability of the union of events under unimodality condition, convexity theory in probabilistic constrained stochastic programming and strong unimodality of multivariate discrete distributions.
We formulate a linear programming problem for bounding the probability of the union of events, where the probability distribution of the occurrences is supposed to be unimodal with known mode and some of the binomial moments of the events are also known. Using a theorem on combinatorial determinants we fully describe the dual feasible bases of a relaxed problem. We present closed form lower and upper bounds for the probability of the union based on two (not necessarily consecutive) as well as first three binomial moments of the random variables involved. We also present upper bounds for the probability of the union based on first four binomial moments. We give a dual method to find customized algorithmic solution of the LP's involved. Numerical examples show that by the use of our bounding methodology, we obtain tighter bounds for the probability of the union.
Next we investigate the convexity theory of programming under probabilistic constraints. Pr'ekopa cite{prekopa73,prekopa95} has proved that if $T$ is an $rtimes n$ random matrix with independent, normally distributed rows such that their covariance matrices are constant multiples of each other, then the function $h(vx)=P(Tvx leq vb)$ is quasi-concave in $R^n$, where $vb$ is a constant vector. We prove that, under same condition, the converse is also true, a special quasi-concavity of $h(vx)$ implies the above-mentioned property of the covariance matrices.
Finally we present sufficient conditions that ensure the strong unimodality of a multivariate discrete distribution and give an algorithm to find the maximum of a strongly unimodal multivariate discrete distribution. We also present examples of strongly unimodal multivariate discrete distributions.
PhysicalDescription
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x, 84 pages
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Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references (p. 77-82).
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Subasi
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Munevver Mine
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Munevver Mine Subasi
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Boros
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Endre Boros
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Prekopa
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Andras
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Andras Prekopa
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Ruszczynski
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Andrej
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Andrej Ruszczynski
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Gurvich
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Vladimir
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Vladimir Gurvich
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Tayfur
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Tayfur Altiok
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Halasz
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Sylvia
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Sylvia Halasz
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Rutgers University
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Graduate School - New Brunswick
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school
OriginInfo
DateCreated (qualifier = exact)
2008
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2008-05
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NjNbRU
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TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Identifier (type = doi)
doi:10.7282/T3ZP46GZ
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
Copyright
Status
Copyright protected
Availability
Status
Open
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Name
Munevver Subasi
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Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
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Non-exclusive ETD license
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Author Agreement 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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