▼ The classical work horse in finance and insurance for modeling asset returns is the Gaussian model. However, when modeling complex random phenomena, more flexible distributions are needed which are beyond the normal distribution. This is because most of the financial and economic data are skewed and have “fat tails” due to the presence of outliers. Hence symmetric distributions like normal or others may not be good choices while modeling these kinds of data. Flexible distributions like skew normal distribution allow robust modeling of high-dimensional multimodal and asymmetric data. In this dissertation, we consider a very flexible financial model to construct robust comonotonic lower convex order bounds in approximating the distribution of the sums of dependent log skew normal random variables. The dependence structure of these random variables is based on a recently developed multivariate skew normal distribution, called unified skew normal distribution. In order to accommodate the distribution to the model so considered, we first study inherent properties of this class of skew normal distribution. These properties along with the convex order and comonotonicity of random variables are then used to approximate the distribution function of terminal wealth. By calculating lower bounds in the convex order sense, we consider approximations that reduce the multivariate randomness to univariate randomness. The approximations are used to calculate the risk measure related to the distribution of terminal wealth. The accurateness of the approximation is investigated numerically. Results obtained from our methods are competitive with a more time consuming method called, Monte Carlo method. The dissertation also includes the study of quadratic forms and their distributions in the unified skew normal setting. Regarding the inferential issue of the distribution, we propose an estimation procedure based on the weighted moments approach. Results of our simulations provide an indication of the accuracy of the proposed method. Advisors/Committee Members: Gupta, Arjun.

▼ In a recent article “Bazzoni's Conjecture” the authors used lattice-theoretic techniques to positively answer a conjecture of Bazzoni's regarding Prüfer domains. Suppose G is an ℓ-group and F is a filter on G+. Recall that F is a principal filter if it is of the form {g ∈ G+ : a ≤ g} for some a ∈ G+;. We say that F is a cold filter if for all P ∈ Min(G), the filter (i.e. the interval) on (G/P)+; defined by FP ={g+P : g ∈ F } has a minimum. If every filter on G+ is principal (resp. cold) then we say that the group is principally-filtered (resp.cold-filtered) ℓ-group. In this dissertation we expand on the ideas of cold filters and characterize cold-filtered ℓ-groups. Advisors/Committee Members: McGovern, Warren.

▼ Ribosomes are protein-synthesizing nano-machines found in all organisms. They are RNA-based enzymes (ribozymes) and among the most ancient nano-assemblies in nature. Recent advances in x-ray crystallography have revealed atomic-resolution structures of a handful of ribosomes from diverse branches of life. Strikingly, although the ribosomal RNA(rRNA) sequences themselves have diverged significantly, the 3D structures of the core regions are highly conserved. We inferred consensus phylogenetic trees using the MrBayes phylogenetics package [19]. We added new modules to FR3D (Find RNA 3D) program suite [43] to take the consensus phylogenetic trees and estimate the mutation rate for each nucleotide in the rRNA conserved cores using dynamic programming described in [11]. We used FR3D to annotate structural features and to determine geometrical relationships between nucleotides belonging to the structurally conserved core regions of the rRNA molecules, as determined in previous work [46]. We report the results of extensive investigation of various explanatory variables, constructed from recurrent structural features observed in the RNA, to model the mutation rates. Of the 45 created variables, calculated for each nucleotide, we identified a parsimonious set of ten that models 36% of the observed variation of mutation rates over 3961 sites. We then performed a factor analysis to determine commonalities of the terms, and hierarchical regression on the main factors to adjust for difference in mutation rate in the 5S, 16S, and 23S chains. Advisors/Committee Members: Zirbel, Craig.

▼ C. D. Bennett, R. Gramlich, C. Hoffman, and S. Shpectorov created Curtis-Tits-Phan theory as a new method for proving the important Phan’s-Theorem and Curtis-Tits’ Theorem. These techniques are a more general method to identify a group, G, as a free amalgamated product of a configuration of low rank subgroups. That is, any group containing such a configuration of subgroups is necessarily a quotient of G. When G is a simple group there is only one quotient that is a non-trivial group and hence any non-trivial group containing the configuration of subgroups must be isomorphic to G. Given compatible presentations of each of the groups in the subgroup configuration, a group presentation can be formed for G. The techniques of Curtis-Tits-Phan theory, at present, have been applied to some groups of Lie-type and Kac-Moody type. However, a specific application of these techniques has not been applied to O±(n,q) and Ω±(n,q) when q is even. In this paper, we applied the techniques of Curtis-Tits-Phan theory to O+(n,q) and Ω+(n,q) when q is even. Specifically,when n ≥ 8 and q ≥ 4 we have identified these groups by free amalgamated products of low rank subgroups. We also identified these groups when n = 6 and q ≥ 8. Furthermore, we have used the GAP computer algebra system to identify the two smallest feasible cases n = 4 and q ∈ {8, 16}. Advisors/Committee Members: Hoffman, Corneliu.

▼ Multiple comparison techniques are to compare more than one pair of treatment means by employing various effective methodologies. In some applications, researchers may believe that before the data are collected, the underlying parameters satisfy an order restriction. But some researchers believe otherwise for certain applications. For instance, in quality control experiments, researchers study the effect of different setting of significant factors, such as the temperature, the pressure, and the different types of machines. The researchers believe that in order to find complete significant results, all treatments should be mixed up and comparisons should be applied to each pair of them. To facilitate multiple comparisons, certain methodologies are proposed and applied. Generally, the frequentist and the Bayesian methodologies are used to conduct multiple comparisons. In this dissertation, we are interested in the Bayesian approach. We propose a hierarchical model in developing and applying multiple comparisons without any restriction in mixed models. The model facilitates inferences in parameterizing the successive differences of the population means, and for them we choose independent prior distributions that are mixtures of a normal distribution and a discrete distribution with its entire mass at zero. For the other parameters, we choose conjugate or non-informative priors, and we derive the full conditional posterior distributions for the parameters in the mixed models. A simulation study is performed to investigate the effectiveness of the proposed hierarchical model. In the simulations, a sequence of different simulated data sets is utilized. To determine the optimal priors for the parameters or hyper-parameters, we do comparisons of the different priors for the purpose of making a decision. In the applications, two real data sets are analyzed to compare the population means for illustrating the performance of the proposed in the hierarchical model. The Type I error, Family-wise Error rate (FWER), and the test power are also studied in the simulations. The simulation results exhibit that the proposed hierarchical model can effectively do multiple comparisons while keeping Type I error relative low and the test power reasonable large in our simulation sets. Gibbs sampler, a special procedure of Markov chain Monte Carlo (MCMC), is used in the simulations to compute the parameters estimates and the posterior probabilities that two population means are equal. The iteration procedure allows one both to determine if any two means are significantly different and to test the homogeneity of all of the means. Hypothesis testing procedure is introduced and the simulations are based on it. The proposed hierarchical model is applied in two real data sets for illustrating the performance of the model. The computing procedure of the proposed hierarchical model can effectively unify parameters estimation, tests of hypotheses, and multiple comparisons in one setting. Advisors/Committee Members: Shang, Junfeng.

▼ Finitely many hypercyclic (respectively, supercyclic) operators acting on a common topological vector space are called disjoint if their direct sum has a hypercyclic (respectively, supercyclic) vector on the diagonal. In this dissertation, we characterize disjointness among hypercyclic and supercyclic linear fractional composition operators on the Hardy space, complementing a celebrated characterization of the cyclic behavior of such operators due to Bourdon and Shapiro [P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (1997)]. We use our characterization to answer a question by Bernal [L. Bernal-Gonzalez, Disjoint hypercyclic operators, Studia Math. 182 Vol 2 (2007) 113-131, Problem 3], whether finitely many hypercyclic composition operators on H(D) generated by non-elliptic automorphisms are disjoint.We also apply our characterization to provide N > 1 invertible hypercyclic operators that are disjoint and so that their inverses are not disjoint supercyclic, solving a problem by Bes and Peris [J. Bes and A. Peris, Disjointness in hypercyclicity, J. Math. Anal. Appl. 336 (2007) 297-315, Problem 3]. We also provide characterizations for disjointness of finitely many hypercyclic (respectively, supercyclic) sequences of composition operators with automorphic symbols of any simply connected domain. We show that finitely many sequences of composition operators induced by automorphic symbols are disjoint hypercyclic if and only if they are disjoint supercyclic, complementing and improving recent work by Bernal, Bonilla, and Calderon [L. Bernal-Gonzalez, A. Bonilla and M. C. Calderon-Moreno, Compositional hypercyclicity equals supercyclicity, Houston Journal of Mathematics 3 No 2 (2007) 581-591]. Finally, we characterize disjointness among powers of supercyclic shift operators on ℓp spaces (1 ≤ p, p finite) complementing the study of the hypercyclic case by Bes and Peris. Advisors/Committee Members: Bes, Juan.

▼ Since the pioneer works of Aigner et. al. (1977) and Meeusen and van den Broeck (1977), stochastic frontier analysis appeared to be a promising field of study for researchers. One of the most important purposes of stochastic frontier analysis is to estimate technical effciencies of firms based on their choices of input combinations, price of inputs and the level of outputs produced. Based on the estimates of technical efficiencies, the rank of firms in terms of efficiency can be obtained. The main idea of stochastic frontier analysis is the introduction of composite error term which contains two components: a technical inefficiency component and a noise component. This composite structure of the error allows each firm to be efficient or inefficient relative to its own production/cost frontier. With the distributional assumptions imposed on the two error terms, the Maximum Likelihood Method or Method of Moments can be employed to obtain estimates for firms' technical efficiencies. Various distributional assumptions have been used in the literature for the two error components. This thesis centered on considering some distributional assumptions for the two error terms. These assumptions are more realistic compared with the existing ones in terms of modeling financial data and allowing for the exibility in the shape of the error terms. The main results obtained were: * Three sets of assumptions including Normal-Uniform, Laplace-Exponential, and Cauchy-Half Cauchy were studied in the context of cross-sectional data. In each model, closed-form formulas for point estimators, lower, and upper limits of technical efficiency were derived. * Each set of assumptions has its own merits in certain circumstances. Flat Uniform distribution is the most neutral one in the sense that one does not impose any prior belief on the distribution of technical inefficiency term relating to its mode or shape. Laplace-Exponential and Cauchy-Half Cauchy are very useful in modeling the financial and economic data, which usually evidence fat tails. * All models were applied to a real data set of 123 U.S electric utility companies. These assumptions generated different estimates of technical efficiency from the four existing models in the literature. * Ranges of skewness coefficients of the distributions of the composite error were compared. Except for the case of Normal-Uniform with zero skewness, the other two proposed models resulted in the wider range of skewness compared to other models used previously, and hence could be used for broader ranges of asymmetry and accommodate outliers. * Two models with distribution sets Normal-Uniform and Cauchy-Half Cauchy were generalized for balanced and unbalanced panel data. Time-invariance technical efficiency models were derived and applied to a balanced panel data set of 683 U.S banks and an unbalanced panel WHO data set. * Allocative efficiency (relating to the ability of a firm in choosing the right combination of inputs given their choices of output levels) was considered besides technical efficiency. These two could be uncorrelated or correlated with each other. In each case, a model was proposed to estimate technical and allocative efficiency. * Lastly, some models to allow for the correlation between the noise components among rms as well as the correlation between the two error components were proposed. In short, the dissertation proposed various models to estimate technical efficiency of firms using either cross-sectional data or panel data. These models have some advantages compared to the existing models. Other contributions included models which account for the correlation among error terms, either among noise components of firms or between noise and technical efficiency terms of firms. Advisors/Committee Members: Gupta, Arjun.

▼ RNA sequence databases contain sequences from hundreds to thousands of homologous molecules. After alignment, these sequences have been successfully mined to reconstruct secondary structures, infer phylogenies, and estimate mutation rates of individual nucleotides and basepairs. Atomic-resolution RNA 3D structures are less numerous, but far more informative than sequence data, as they show, in the ideal case of well-ordered and well-resolved structures, every basepair, base stacking, and base-backbone hydrogen bond. To function correctly, structured RNA molecules must fold into the correct 3D structure. Since RNA 3D structure is more conserved than RNA primary sequence, detecting structural similarities (and dissimilarities) among RNA 3D structures can produce a wealth of information regarding the functional and evolutionary properties that could not be found by analyzing sequence data alone. The goal of RNA 3D structural alignment is to establish correspondences between the individual nucleotides that are similar in the two 3D structures. Due to rapid technological developments within the past decade, there has been a dramatic increase in both the size and number of RNA 3D structures that have been crystallized and made available. It is no longer feasible to solely rely on manual comparison of two RNA 3D structures, which can be a labor-intensive and time-consuming process. Consequently, it has become essential to develop tools that are capable of accurately and automatically discovering structural similarities among RNA molecules, which is accomplished through 3D-to-3D alignment. Advisors/Committee Members: Zirbel, Craig.

▼ In this dissertation we investigated the two topics hypercyclicity and the Aluthge transform. Each of these is related to the Invariant Subspace Problem. On a topological vector space X, a linear operator T:X → X is said to be hypercyclic if there is a vector x for which the sequence x,Tx, T2x,T3x,... is dense in X. We explored whether the dynamical properties of an operator are preserved by the Aluthge transform. We showed for bilateral weighted shifts, an operator T is mixing, chaotic, or hypercyclic if and only if the Aluthge transform of the operator has the same dynamical property. We also supplied conditions for when the Aluthge transform of an arbitrary operator T has the same dynamical properties as T. In chapter three we provided a strong counterexample to a conjecture by Jung, Ko, and Percy. They conjectured that for every bounded linear operator T on a Hilbert space, the sequence of operators formed by iteratively applying the Aluthge transform to T would converge to a normal operator. We used a probabilistic argument to show that if T is any bilateral forward shift, then either the sequence of iterations of the Aluthge transform converges to a normal shift in the strong operator topology, or it fails to converge in a dramatic sense in that its set of strong operator topology subsequential limits is an “interval” of normal shifts. We then showed for any positive reals a < b, there is a bilateral weighted forward shift T for which the set of strong operator topology subsequential limits of the sequence of iterates of the Aluthge transform is the set of shifts of the form tS where S is the pure forward shift, and t is any number in the interval [a,b]. These results were extended to address complexly weighted shifts, and bilateral backward shifts. In the last chapter, we address where “most” hypercyclic vectors are located relative to the range of a hypercyclic operator. If x is hypercyclic for T, then so is Tnx for every natural number n, and Tn x is in the range of T. Since moreover, the range of T is dense in X, one might expect that most if not all of an operators hypercyclic vectors lie in its range. To the contrary, we showed for every non-surjective hypercyclic operator T on a Banach space,the set of hypercyclic vectors for T that are not in its range is large in that it is a set of category II. We also provided a sense by which the range of an arbitrary hypercyclic operator is large in its set of hypercyclic vectors for T. Advisors/Committee Members: Juan, Juan.

▼ Obtaining reasonable error estimates for computed solutions to hyperbolic conservation laws using the Runge-Kutta / Discontinuous Galerkin Method (RKDG) has been an issue for many years. Zhang and Shu have written papers dealing with an a priori estimate for a third order Runge-Kutta scheme. In this dissertation a new way to compute an a posteriori error estimate for numerical solutions computed using RKDG methods is provided. These techniques can be applied to any order Runge-Kutta scheme as well as any order polynomial. The idea of smoothness indicators is discussed in the papers by Sun, Sun and Fillippova, and Rumsey and Sun. These smoothness indicators are used to compute an a posteriori error estimate. In providing the framework for this new technique, only the regions where the solution is smooth are considered. Two examples, Burgers' equation and the traffic law, are provided to give the full details of the error analysis and to show that the errors computed are reasonable. In the final remarks some insight into how to adapt this method for regions where the solution has a fully developed shock or a partially developed shock are discussed. Advisors/Committee Members: Sun, Tong.

▼ Hypercyclicity is the study of linear and continuous operators that possess a dense orbit. Given a separable, infinite dimensional topological vector space X, we say a continuous linear operator T is hypercyclic if there exists a vector x in X such that its orbit Orb(T,x)={x, Tx, T²x, …} is dense in X. Many interesting phenomena appear when analyzing the behavior of iterates of linear and continuous operators, in particular we emphasize the existence of several zero-one laws. We first note that, if an operator T has a hypercyclic vector, it has a dense Gδ set of such vectors, and hence the set of hypercyclic vectors for an operator is either empty or very large in a topological sense. Next, by proving that a somewhere dense orbit is everywhere dense, P. S. Bourdon and N. S. Feldman showed a second zero-one law which states that either an orbit Orb(T,x) is nowhere dense or it is dense in the whole space. In my dissertation we uncovered the existence of another such zero-one law for certain classes of operators. We showed that for a weighted backward shift on ℓp to be hypercyclic it suffices to require the operator to have an orbit Orb(T,x) with a single non-zero limit point, thus relaxing Bourdon and Feldman’s condition of having a dense orbit in some open subset of X. However, our condition does not guarantee that the original orbit Orb(T,x) is dense in X, nonetheless we can demonstrate how to construct a hypercyclic vector for T by using the non-zero limit point of the orbit. Even more interestingly, the condition above can be relaxed to simply requiring that the orbit has infinitely many members in a ball whose closure avoids the zero vector. To summarize this behavior of weighted backward shifts, we emphasize that a shift T is not hypercyclic if and only if every set of the form Orb(T,x)∪{0} is closed in ℓp . Thus we showed the existence of a zero-one law for the hypercyclicity of these shifts, which states that either no orbit has a non-zero limit point in ℓp; or some orbit has every vector in ℓp as a limit point. Furthermore we showed that this zero-one law for the hypercyclic behavior of shifts is also shared by other classes of operators, in particular the adjoints of the multiplication operators on the Bergman space A2(Ω) for an arbitrary region Ω. To achieve this we cannot borrow techniques used for the shift operators, but instead we have to take a function theoretical approach. However, we also showed that this behavior does not generalize to all classes of operators, namely we provided an example of a linear fractional composition operator on the Hardy space H2(D) that is not hypercyclic, and yet it has an orbit with a non-constant limit point. To summarize the importance of our results, we would like to point out that in our endeavor to study the phenomena of hypercyclicity it is important to understand how an operator fails to be hypercyclic. Having proved that for certain classes of operators, a non-hypercyclic operator can at most have the zero vector as an orbital limit point, we have shown that these operators fail at having a dense orbit in quite a dramatic way. Thus we described the hypercyclic behavior of certain operators as a zero-one law of orbital limit points, and so we have uncovered another facet of hypercyclicity associated with dichotomous behavior. Advisors/Committee Members: Chan, Kit.

▼ The problem of finding the minimum effective dose (MED) of a drug involves multiple comparisons. In such a problem, generally, family-wise error rate is inflated if multiplicity adjustment is not made appropriately. The sample sizes are also an important issue that we need to consider from the aspect of efficiency, such as costs in studies. In this dissertation, we propose several novel procedures for MED under heteroscedasticity, which control family-wise error rate as well as the accuracy level for identifying MED. In the literature, Bonferroni adjustment is widely used due to it's simplicity, but it is too conservative. Hsu and Berger (1999) proposed a stepwise confidence intervals procedure for MED using partitioning principle, which controlled the family-wise error rate without multiplicity adjustment. Their method is based on the cases of equal variance assumption. However, equal variance assumption is seldom satisfied in practice. Tao et al. (2002) extended Hsu and Berger's procedure for MED under heteroscedasticity. When variances in different dose groups are not equal, to determine the minimum effective dose, we have to deal with the Behrens-Fisher problem (comparing means from two normal populations when population variances cannot be assumed equal). Exact solutions to Behrens-Fisher problem can be obtained by using two-stage procedures. For instance, Chapman (1950) extended Stein's (1945) two-stage sampling procedure in two-sample case. In this dissertation, we examine methods in the comparisons of two populations and innovatively construct three new stepwise procedures that shed new light in the aspect of asymptotic efficiency, power improvement, and accuracy control in the inference of the minimum effective dose of a drug. Simulation studies greatly enhance and confirm the desired theoretical results. The procedures are applied to the analysis of the pharmacologic effect of an experimental compound on relative organ weights in mice. Advisors/Committee Members: Chen, John.

▼ In this study we create smoothness indicators which quantitively measure the smoothness of numerical approximations to solutions of nonlinear conservation laws. We perform numerical tests for these smoothness indicators applied to numerical approximations obtained by applying ENO and WENO schemes. Based on our numerical results, we believe that first these smoothness indicators can measure the numerical smoothness of numerical solutions, and second these smoothness indicators can be used to compare the smoothness of various ENO and WENO schemes. Thus, we believe that these indicators will be useful for the error analysis of ENO and WENO schemes for nonlinear conservation laws. Advisors/Committee Members: Sun, Tong.

▼ The family of stable distributions is an important class of four-parameter continuous distributions. It has appealing properties such as being the only possible limit distribution of a suitably normalized sum of independent and identically distributed random variables. Therefore, it has wide applications in modeling distributions with heavy tails, such as the return of financial assets. However, there are also critics against using stable distribution in modeling financial assets such as stock and futures. It is very important to check the validity of the model assumption before making inferences based on the model. Previous work has been done in the goodness-of-fit test for several special cases including normal distributions, Cauchy distributions and more generally, symmetric stable distributions. Classical goodness-of-fit methods such as the Kolmogorov-Smirnov test and the Anderson-Darling test are not able to handle the stable distributions directly because of the lack of closed-form probability density functions (PDF) and cumulative distribution functions (CDF). Since stable distributions can be fully characterized by their characteristic functions, goodness-of-fit tests based on the empirical characteristic function (ECF) have also been studied in recent years. In this dissertation, a new goodness-of-fit test is proposed for general stable distributions based on the energy statistic, which is invariant under rigid motions. The test statistic is essentially a weighted $L^2$-norm of the distance between the empirical characteristic function and the hypothetical characteristic function of the null distribution, and it can also be expressed as a V-statistic with degenerate kernel. By asymptotic theory of degenerate kernel V-statistics, the test statistic converges in distribution to an infinite sum of weighted chi-square random variables if the null hypothesis of stability is true. It can be proved that the test is consistent against a large class of alternatives. A relatively simple computation formula is derived for the test statistic, which involves numerical integration in general. Bootstrap method and critical values based on the asymptotic distribution of the test statistic can be applied to implement the test. The dissertation is organized as follows. In Chapter 1, the class of stable distributions and its properties are reviewed. In Chapter 2, existing methods of goodness-of-fit test for stable distributions will be discussed. In Chapter 3, theoretical properties of the test statistic, including the definition, computation issues and asymptotic results, are developed. In Chapter 4, simulation studies are presented to illustrate the empirical type I error and power of testing stable distributions against alternative distributions including stable distributions with different parameters and other interesting light-tailed and heavy-tailed distributions. Simulation results show that our test is sensitive in detecting the difference either in the center or extreme values in the tail. In Chapter 5, some basic work has been finished to study the asymptotic distribution of the energy statistic for testing Cauchy when parameters are estimated by maximum likelihood estimators (MLE). Advisors/Committee Members: Rizzo, Maria L.