Mathematics Theses and Dissertations
http://hdl.handle.net/1903/2793
2017-11-25T02:00:12ZNonparametric Estimation of a Distribution Function in Biased Sampling Models
http://hdl.handle.net/1903/20187
Nonparametric Estimation of a Distribution Function in Biased Sampling Models
Xu, Jian-Lun
The nonparametric maximum likelihood estimator (NPMLE) of a distribution function F in biased sampling models have been studied by Cox (1969), Vardi (1982, 1985), and Gill, Vardi, and Wellner (1988). Their approaches are based on the assumption that the observations are drawn from biased distributions of F and biasing functions do not depend on F. These assumptions have been used in Patil and Rao (1978). This thesis extends the biased sampling model by making the biasing functions depend on the distribution function F in a variety of ways. With this extension, many of the existing models, including the ranked-set sampling model and the nomination sampling model, become special cases of the biased sampling model. The statistical inference about F becomes to a large extent the study of the biasing function. We develop conditions under which the generalized model is identifiable. Under these conditions, an estimator of the underlying distribution F is proposed and its strong consistency and asymptotic normality are established. In certain situation, estimation of Fin a biased sampling model is in fact a problem of estimating a monotone decreasing density. Several density estimators are studied. They include the nonparametric maximum likelihood estimator, a kernel estimator, and a modified histogram type estimator. The strong consistency, the asymptotic normality, and the bounds on average error for the estimators are studied in detail. In summary, this thesis is a generalizations of the estimation results available for the ordinary s-biased sampling model, the ranked-set sampling model, the nomination sampling model, and a monotone decreasing density.
1993-01-01T00:00:00ZApplication of Mathematical and Computational Models to Mitigate the Overutilization of Healthcare Systems
http://hdl.handle.net/1903/20003
Application of Mathematical and Computational Models to Mitigate the Overutilization of Healthcare Systems
Hu, Xia
The overutilization of the healthcare system has been a significant issue financially and politically, placing burdens on the government, patients, providers and individual payers. In this dissertation, we study how mathematical models and computational models can be utilized to support healthcare decision-making and generate effective interventions for healthcare overcrowding. We focus on applying operations research and data mining methods to mitigate the overutilization of emergency department and inpatient services in four scenarios. Firstly, we systematically review research articles that apply analytical queueing models to the study of the emergency department, with an additional focus on comparing simulation models with queueing models when applied to similar research questions. Secondly, we present an agent-based simulation model of epidemic and bioterrorism transmission, and develop a prediction scheme to differentiate the simulated transmission patterns during the initial stage of the event. Thirdly, we develop a machine learning framework for effectively selecting enrollees for case management based on Medicaid claims data, and demonstrate the importance of enrolling current infrequent users whose utilization of emergency visits might increase significantly in the future. Lastly, we study the role of temporal features in predicting future health outcomes for diabetes patients, and identify the levels to which the aggregation can be most informative.
2017-01-01T00:00:00ZParabolic Higgs bundles and the Deligne-Simpson Problem for loxodromic conjugacy classes in PU(n,1)
http://hdl.handle.net/1903/19998
Parabolic Higgs bundles and the Deligne-Simpson Problem for loxodromic conjugacy classes in PU(n,1)
Maschal Jr, Robert Allan
In this thesis we study the Deligne-Simpson problem of finding matrices $A_j\in C_j$ such that $A_1A_2\ldots A_k = I$ for $k\geq 3$ fixed loxodromic conjugacy classes $C_1,\ldots,C_k$ in $PU(n,1)$. Solutions to this problem are equivalent to representations of the $k$ punctured sphere into $PU(n,1)$, where the monodromy around the punctures are in the $C_j$. By Simpson's correspondence \cite{s1}, irreducible such representations correspond to stable parabolic $U(n,1)$-Higgs bundles of parabolic degree 0. A parabolic $U(n,1)$-Higgs bundle can be decomposed into a parabolic $U(1,1)$-Higgs bundle and a $U(n-1)$ bundle by quotienting out by the rank $n-1$ kernel of the Higgs field. In the case that the $U(1,1)$-Higgs bundle is of loxodromic type, this construction can be reversed, with the added consequence that the stability conditions of the resulting $U(n,1)$-Higgs bundle are determined only by the kernel of $\Phi$, the number of marked points, and the degree of the $U(1,1)$-Higgs bundle. With this result, we prove our main theorem, which says that when the log eigenvalues of lifts $\widetilde{C}_j$ of the $C_j$ to $U(n,1)$ satisfy the inequalities in \cite{biswas} for the existence of a stable parabolic bundle, then there is a stable parabolic $U(n,1)$-Higgs bundle whose monodromies around the marked points are in $\widetilde{C}_j$. This new approach using Higgs bundle techniques generalizes the result of Falbel and Wentworth in \cite{fw1} for fixed loxodromic conjugacy classes in $PU(2,1)$.
This new result gives sufficient, but not necessary, conditions for the existence of an irreducible solution to the Deligne-Simpson problem for fixed loxodromic conjugacy classes in $PU(n,1)$. The stability assumption cannot be dropped from our proof since no universal characterization of unstable bundles exists. In the last chapter, we explore what happens when we change the weights of the stable kernel in the special case of three fixed loxodromic conjugacy classes in $PU(3,1)$. Using the techniques from \cite{fw2}, \cite{fw1}, and \cite{paupert}, we can show that our construction implies the existence of many other solutions to the problem.
2017-01-01T00:00:00ZDESCRIBING URGENT EVENT DIFFUSION ON TWITTER USING NETWORK STATISTICS
http://hdl.handle.net/1903/19997
DESCRIBING URGENT EVENT DIFFUSION ON TWITTER USING NETWORK STATISTICS
Sun, Hechao
In this dissertation, I develop a novel framework to study the diffusion of urgent events through the popular social media platformâ€”Twitter. Based on my literature review, this is the first comprehensive study on urgent event diffusion through Twitter. I observe similar diffusion patterns among different data sets and adopt the "cross prediction" mode to handle the early time prediction problem. I show that the statistics from the network of Twitter retweets can not only provide profound insights about event diffusion, but also can be used to effectively predict user influence and topic popularity. The above findings are consistent across various experiment settings. I also demonstrate that linear models consistently outperform state-of-art nonlinear ones in both user and hashtag prediction tasks, possibly implying the strong log-linear relationship between selected prediction features and the responses, which potentially could be a general phenomenon in the case of urgent event diffusion.
2017-01-01T00:00:00Z