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Two parametric models for income and financial return distributions are presented. There are the four-parameter normal Laplace (NL) and the five-parameter generalized normal Laplace (GNL) distributions. Their properties are discussed; furthermore, estimation of the parameters by the method of moments and maximum likelihood is presented. The performances of fitting the two models to nine empirical distributions of family income have been evaluated and compared against the four- and five-parameter generalized beta2 (GB2) and generalized beta (GB) distributions which had been previously claimed as best-fitting four- and five- parameter models for income distribution. The results demonstrate that the NL distribution has better performance than the GB and GB2 distributions with the GNL distribution providing an even better fit. Limited application to data on financial log returns shows that the fit of the GNL is comparable to the well-known generalized hyperbolic distribution. However, the GNL suffers from a lack of closed-form expressions for its probability density and cumulative distribution functions, and fitting the distribution numerically is slow and not always reliable. The results of this thesis suggest a strong case for considering the GNL family as parametric models for income data and possibly for financial logarithmic returns.

This thesis involves the study of a repulsive-attractive N-body problem, which is a subclass of a quasihomogeneous N-body problem [5]. The quasihomogeneous N-body problem is the study of N point masses moving in R3N, where the negative of the potential energy is of the form, X 1≤i<j≤N bmimjr−β ij + X 1≤i<j≤N amimjr−α ij . In the above equation, rij is the distance between the point mass mi and the point mass mj , and a, b, α > β > 0 are constants. The repulsive-attractive N-body problem is the case where a < 0 and b > 0. We start the ground work for the study of the repulsive-attractive N-body problem by defining the first integrals, collisions and pseudo-collisions and the collision set. By examining the potentials where a < 0 and b > 0, we see that the dominant force is repulsive. This means that the closer two point masses get the greater the force acting to separate them becomes. This property leads to the main result of the first chapter: there can be no collisions or pseudo-collisions for any repulsive-attractive system. In the next chapter we study central configurations of the system. Quasihomogeneous potentials will have different central configurations than homogeneous potentials [6], thus requiring the classification of two new subsets of central configurations. Loosely speaking, the set of central configurations that are not central configurations for any homogeneous potential are called extraneous. The set of configurations that are central configurations for both homogeneous potentials that make up the quasihomogeneous potential, are called simultaneous configurations. We also notice that every simultaneous central configuration will be non-extraneous, therefore the two subsets are disjoint. Next we show the existence of oscillating homothetic periodic orbits associated with non-extraneous configurations. Finally in this chapter, we investigate the polygon solutions for repulsive-attractive N-body problems [11]. In particular we show that the masses need no longer to be equal, for repulsive-attractive potentials. It will be shown that there exists a square configuration with m1 = m2 6= m3 = m4, that leads to a relative equilibrium. Therefore, for N = 4 the set of extraneous configurations is non-empty. The last chapter deals with the complete analysis of the generalized Lennard- Jones 2-body problem. The generalized Lennard-Jones problem is the subcase of the repulsive-attractive N-body problem, where a = −1, b = 2, and α = 2β. We proceed as in [13] by using diffeomorphic transforms to get an associated system thereby generating a picture of the global flow of the system. This gives us the complete flow for the generalized Lennard-Jones 2-body problem.

Correspondence analysis is a descriptive and explorative technique for the study of associations between variables. It is a visualization method for analyzing high dimensional data via projections onto a low-dimensional subspace. In this thesis, we demonstrate the applicability of correspondence analysis to microarray data. We show that it can be used to identify important genes and treatment patterns by coordinating and projecting the genes and the experimental conditions. In addition, we estimate missing values in the gene expressions using the Expectation-Maximization (EM) algorithm and identify genes with large between-condition variability using the projections of the genes and the conditions. To demonstrate its application, correspondence analysis is applied to various simulated data and microarray data from the EPA (Environmental Protection Agency) studies. We conclude that correspondence analysis is a useful tool for analyzing the associations between genes and experimental conditions, for identifying important genes, and for estimating missing values.

In this thesis study, three different feature selection methods, LASSO, SLR, and SMLR, were tested and compared using microarray fold change data. Two real datasets were used to first investigate and compare the ability of the algorithms in selecting feature genes on data under two conditions. It was found that SMLR was quite sensitive to its parameter, and was more accurate in selecting differentially expressed genes when compared to SLR and LASSO. In addition, the model coefficients generated by SMLR had a close relationship with the magnitude of fold changes. Also, SMLR's ability in selecting differentially expressed genes with data that had more than two conditions was shown to be successful. The results from simulation experiments agreed with the results from the real dataset experiments. Additionally, it was found that different proportions of differentially expressed genes in the data did not affect the performance of LASSO and SLR, but the number of genes selected by SMLR increased with the proportion of regulated genes. Also, as the number of replicates used to build the model increased, the number of genes selected by SMLR increased. This applied to both correctly and incorrectly selected genes. Furthermore, it was found that SMLR performed the best in identifying future treatment samples.

A permutation array, represented by PA(n, d), is a subset of Sn such that any two distinct elements have a distance of at least d where d is the number of differing positions. We analyze the upper and lower bounds of permutation codes with distance equal to 4. An optimization problem on Young diagrams is used to improve the upper bound for almost all n while the lower bound is improved for small values of n by means of recursive construction methods.

A k-net of order n is an incidence structure consisting of n2 points and nk lines. Two lines are said to be parallel if they do not intersect. A k-net of order n satisfies the following four axioms: (i) every line contains n points; (ii) parallelism is an equivalence relation on the set of lines; (iii) there are k parallel classes, each consisting of n lines and (iv) any two non-parallel lines meet exactly once. A Latin square of order n is an n by n array of symbols in which each row and column contains each symbol exactly once. Two Latin squares L and M are said to be orthogonal if the n2 ordered pairs (Li,j , Mi,j ) are all distinct. A set of t mutual ly orthogonal Latin squares is a collection of Latin squares, necessarily of the same order, that are pairwise orthogonal. A k-net of order n is combinatorially equivalent to k − 2 mutually orthogonal Latin squares of order n. It is this equivalence that motivates much of the work in this thesis. One of the most important open questions in the study of Latin squares is: given an order n what is the maximum number of mutually orthogonal Latin squares of that order? This is a particularly interesting question when n is congruent to two modulo four. A code is constructed from a net by defining the characteristic vectors of lines to be generators of the code over the finite field F2 . Codes allow the structure of nets to be profitably explored using techniques from linear algebra. In this dissertation a framework is developed to study linear dependence in the code of the net N6 of order ten. A complete classification and combinatorial description of such dependencies is given. This classification could facilitate a computer search for a net or could be used in conjunction with more refined techniques to rule out the existence of these nets combinatorially. In more generality relations in 4-nets of order congruent to two modulo four are also characterized. One type of dependency determined algebraically is shown not to be combinatorially feasible in a net N6 of order ten. Some dependencies are shown to be related geometrically, allowing for a concise classification. Using a modification of the dimension argument first introduced by Dougherty [19] new upper bounds are established on the dimension of codes of nets of order congruent to two modulo four. New lower bounds on some of these dimensions are found using a combinatorial argument. Certain constraints on the dimension of a code of a net are shown to imply the existence of specific combinatorial structures in the net. The problem of packing points into lines in a prescribed way is related to packing problems in graphs and more general packing problems in combinatorics. This dissertation exploits the geometry of nets and symmetry of complete multipartite graphs and combinatorial designs to further unify these concepts in the context of the problems studied here.

A generalization of the circular chromatic number to hypergraphs is devel-oped. Circular colourings of graphs and hypergraphs are first discussed and it is shown that the circular chromatic number of a graph is the same regard-less of whether the hypergraph or graph definition is used. After presenting a few basic results, some examples of circular chromatic numbers of various families of hypergraphs are given. Subsequently, the concepts of the star chromatic number and the arc chromatic number are introduced. Specif¬ically, both numbers are shown to be equivalent to the circular chromatic number. Finally the relationship between the imbalance of a hypergraph and the circular chromatic number is explored and a classical result of Minty is deduced.

A k-uniform hypergraph N = (X. B) of order n is a family of k-subsets B of an n-set X. A k-uniform hypergraph 7--L = (X. B) is disjoint union-free (DUF) if all disjoint pairs of elements of B have distinct unions; that is, if for every A, B, C, D E B. A fl B = C f1 D = 0 and A U B =CUD implies {A. B} = {C, D}. DUF families of maximum size have been studied by Erdos and Fiiredi. and in the case k = 3 this maximum size has been conjectured to equal (z). In this thesis, we study DUF 3-uniform hypergraphs with the main goals of presenting evidence to support this conjecture and studying the structures that have conjectured maximum size. If each pair of distinct elements of X is covered exactly A times in B then we call N = (X, B) an (n. k. A)-design. Using a blend of graph- and design-theoretic techniques, we study the DUF (n,. 3. 3)-designs that are the conjectured unique structures having maximum size. Central results of this thesis include substantially improving lower bounds on the maximum size for a large class of n. giving conditions on pair coverage in a DUF 3-uniform hypergraph that force an (n., 3, 3)-design, and providing constructions for DUF 3-uniform hypergraphs from families of DUF hypergraphs with smaller orders. Let. N = (X, B) be a DUF k-uniform hypergraph with the property that 7-t U {E} is not DUF for any k-subset E of X not already in H. Then N is maximally DUF. We introduce the problem of finding the minimum size of maximally DUF families and provide bounds on this quantity for k = 3.

Robust statistics is an extension of classical parametric statistics, which provides a safeguard against gross errors in experiments. Effectively, robustness properties of Uhlig's Q-estimators are examined and compared with that. of Rocke's Ai-estimators. In particular, the finite-sample implosion and explosion breakdown points are inves-tigated and introduced into constructing robust designs for the one-way random effects model. Optimal robust designs based on Uhlig's Q-estimation are similar to the ones based on Rocke's M-estimation. Ultimately. robust estimation procedures would provide steady and reliable estimates of model parameters in case of the occurrence of outliers.