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TitleNew results in probability bounding, a convexity statement and unimodality of multivariate discrete distributions
NameSubasi, Munevver Mine (author), Boros, Endre (chair), Prekopa, Andras (internal member), Ruszczynski, Andrej (internal member), Gurvich, Vladimir (internal member), Altiok, Tayfur (internal member), Halasz, Sylvia (outside member), Rutgers University, Graduate School - New Brunswick,
Degree Date2008-05
Date Created2008
SubjectOperations Research,
Distribution (Probability theory),
Multivariate analysis
DescriptionThis 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.
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.
NotePh.D.
NoteIncludes bibliographical references (p. 77-82).
Genretheses
Persistent URLhttp://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.17395
LanguageEnglish
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization Name
RightsThe author owns the copyright to this work.