This thesis examines the labor force participation decisions of married couples, and special attention is paid to a spouse’s health conditions affecting their own and the spouse’s labor force participation decision. I used the Health and Retirement Study survey data and estimated a seemingly unrelated bivariate probit model. A number of variables besides health condition were added: age, education level, and family unearned income. The results of this research paper support the findings from the relevant literature that the labor supply decisions of the husband and wife are related. The oldest age group is least likely to work. The younger the husband, the more likely it is that the husband will work. At the ages between 40 and 49, wives have the biggest probability to work. The higher the education level, the more likely it is that a spouse is going to work. The more total family unearned income, the less probable the spouse will go to work. Poor health has a negative effect on labor force participation and a positive effect for the spouse’s labor force participation.

Splines are piecewise polynomial functions of a given order of smoothness r. Given complex delta the set of splines of degree less than or equal to d forms a vector space and is denoted by Sr d(delta). For a simplicial complex delta, Strang conjectured a lower bound on the dimension of spline space Srd(delta) and it is known that the equality holds for sufficiently large d. It is called the dimension formula. In this dissertation, we approach the study of splines from the viewpoint of algebraic geometry. This dissertation follows the works of Lau and Stiller. They introduced the conformality conditions which lead to the machinery of sheaves and cohomology which provided a powerful type of generalization of linear algebra. First, we try to analyze effects in the dimensions of spline spaces when we remove or add certain faces in the given complex. We define the cofactor spaces and cofactor maps from the given complexes and use them to interpret the changes in the dimensions of spline spaces. Second, given polyhedral complex delta, we break it into two smaller complexes delta1 and delta2 which are usually easier to handle. We will find conditions for delta1 and delta2 which guarantee that the dimension formula holds for the original complex delta. Next, we use the previous splitting method on certain types of triangulations. We explain how to break the given triangulation and show what kind of simple complexes we end up with. Finally, we study the "2r+1" conjecture on a certain triangulation. The "2r+1" conjecture is that the dimension formula holds on any triangulation for d >/= 2r + 1. We know that the conjecture is sharp because the dimension formula fails on a certain triangulation for d = 2r, but we do not know if it holds on the same triangulation when d = 2r + 1. It is related to a Toeplitz matrix.

In the field of spatial statistics, it is often desirable to generate a smooth surface for a region over which only noisy observations of the surface are available at some locations, or even across time. Kriging and kernel estimations are two of the most popular methods. However, these two methods become problematic when the domain is not regular, such as when it is rectangular or convex. Bivariate B-splines developed by mathematicians provide a useful nonparametric tool in bivariate surface modeling. They inherit several appealing properties of univariate B-splines and are applicable in various modeling problems. More importantly, bivariate B-splines have advantages over kriging and kernel estimation when dealing with complicated domains. The purpose of this dissertation is to develop a nonparametric surface fitting method by using bivariate B-splines that can handle complex spatial domains. The dissertation consists of four parts. The first part of this dissertation explains the challenges of smoothing over complicated domains and reviews existing methods. The second part introduces bivariate B-splines and explains its properties and implementation techniques. The third and fourth parts discuss application of the bivariate B-splines in two nonparametric spatial surface fitting problems. In particular, the third part develops a penalized B-splines method to reconstruct a smooth surface from noisy observations. A numerical algorithm is derived, implemented, and applied to simulated and real data. The fourth part develops a reduced rank mixed-effects model for functional principal components analysis of sparsely observed spatial data. A numerical algorithm is used to implement the method and tested on simulated and real data.

The introduction of new schemes that are based on the communication among nodes has motivated the use of composite fading models due to the fact that the nodes experience different multipath fading and shadowing statistics, which subsequently determines the required statistics for the performance analysis of different transceivers. The end-to-end signal-to-noise-ratio (SNR) statistics plays an essential role in the determination of the performance of cascaded digital communication systems. In this thesis, a closed-form expression for the probability density function (PDF) of the end-end SNR for independent but not necessarily identically distributed (i.n.i.d.) cascaded generalized-K (GK) composite fading channels is derived. The developed PDF expression in terms of the Meijer-G function allows the derivation of subsequent performance metrics, applicable to different modulation schemes, including outage probability, bit error rate for coherent as well as non-coherent systems, and average channel capacity that provides insights into the performance of a digital communication system operating in N cascaded GK composite fading environment. Another line of research that was motivated by the introduction of composite fading channels is the error performance. Error performance is one of the main performance measures and derivation of its closed-form expression has proved to be quite involved for certain systems. Hence, in this thesis, a unified closed-form expression, applicable to different binary modulation schemes, for the bit error rate of dual-branch selection diversity based systems undergoing i.n.i.d. GK fading is derived in terms of the extended generalized bivariate Meijer G-function.

Scattered data collected at sample points may be used to determine simple functions to best fit the data. An ideal choice for these simple functions is bivariate splines. Triangulation of the sample points creates partitions over which the bivariate splines may be defined. But the optimality of the approximation is dependent on the choice of triangulation. An algorithm, referred to as an Edge Swapping Algorithm, has been developed to transform an arbitrary triangulation of the sample points into an optimal triangulation for representation of the scattered data. A Matlab package has been completed that implements this algorithm for any triangulation on a given set of sample points.

The univariate and bivariate central chi-square- and F distributions have received a decent amount of attention in the literature during the past few decades; the noncentral counterparts of these distributions have been much less present. This study enriches the existing literature by proposing bivariate noncentral chi-square and F distributions via the employment of the compounding method with Poisson probabilities. This method has been used to a limited extent in the field of distribution theory to obtain univariate noncentral distributions; this study extends some results in literature to the corresponding bivariate setting. The process which is followed to obtain such bivariate noncentral distributions is systematically described and motivated. Some distributions of composites (univariate functions of the dependent components of the bivariate distributions) are derived and studied, in particular the product, ratio, and proportion. The benefit of introducing these bivariate noncentral distributions and their respective composites is demonstrated by graphical representations of their probability density functions. Furthermore, an example of possible application is given and discussed to illustrate the versatility of the proposed models. / Dissertation (MSc)--University of Pretoria, 2014. / Statistics / MSc / Unrestricted

The purpose of this thesis is to compare rank-based tests of bivariate stochastic order. Given two bivariate distributions $F$ and $G$, the general problem we are dealing with is to test $H_0: F=G$ against $H_1:F<G$, where $F$ and $G$ are independent continuous distributions on $\Re ^2$. (``$F<G$" means that $F(x)\leq G(x)~\forall x\in \Re^2$, and $\exists x\in \Re^2$ such that $F(x)< G(x)$.). In particular, we will analyze three analogues of the one-dimensional Mann-Whitney-Wilcoxon test in two dimensions. Two of the test statistics are new; we call them the Kendall and Spearman statistics. We will then show the asymptotic distributions and carry out empirical comparisons of the Kendall, Spearman and the third two-dimensional Mann-Whitney-Wilcoxon statistics.

The purpose of this thesis is to compare rank-based tests of bivariate stochastic order. Given two bivariate distributions $F$ and $G$, the general problem we are dealing with is to test $H_0: F=G$ against $H_1:F<G$, where $F$ and $G$ are independent continuous distributions on $\Re ^2$. (``$F<G$" means that $F(x)\leq G(x)~\forall x\in \Re^2$, and $\exists x\in \Re^2$ such that $F(x)< G(x)$.). In particular, we will analyze three analogues of the one-dimensional Mann-Whitney-Wilcoxon test in two dimensions. Two of the test statistics are new; we call them the Kendall and Spearman statistics. We will then show the asymptotic distributions and carry out empirical comparisons of the Kendall, Spearman and the third two-dimensional Mann-Whitney-Wilcoxon statistics.

The purpose of this thesis is to compare rank-based tests of bivariate stochastic order. Given two bivariate distributions $F$ and $G$, the general problem we are dealing with is to test $H_0: F=G$ against $H_1:F<G$, where $F$ and $G$ are independent continuous distributions on $\Re ^2$. (``$F<G$" means that $F(x)\leq G(x)~\forall x\in \Re^2$, and $\exists x\in \Re^2$ such that $F(x)< G(x)$.). In particular, we will analyze three analogues of the one-dimensional Mann-Whitney-Wilcoxon test in two dimensions. Two of the test statistics are new; we call them the Kendall and Spearman statistics. We will then show the asymptotic distributions and carry out empirical comparisons of the Kendall, Spearman and the third two-dimensional Mann-Whitney-Wilcoxon statistics.

The purpose of this thesis is to compare rank-based tests of bivariate stochastic order. Given two bivariate distributions $F$ and $G$, the general problem we are dealing with is to test $H_0: F=G$ against $H_1:F<G$, where $F$ and $G$ are independent continuous distributions on $\Re ^2$. (``$F<G$" means that $F(x)\leq G(x)~\forall x\in \Re^2$, and $\exists x\in \Re^2$ such that $F(x)< G(x)$.). In particular, we will analyze three analogues of the one-dimensional Mann-Whitney-Wilcoxon test in two dimensions. Two of the test statistics are new; we call them the Kendall and Spearman statistics. We will then show the asymptotic distributions and carry out empirical comparisons of the Kendall, Spearman and the third two-dimensional Mann-Whitney-Wilcoxon statistics.