In the representation of scattered data by smooth pp (:= piecewise polynomial) functions, perhaps the most important problem is to find an optimal triangulation of the given sample sites (called vertices). Of course, the notion of optimality depends on the desirable properties in the approximation or modeling problems. In this paper, we are concerned with optimal approximation order with respect to the given order r of smoothness and degree k of the polynomial pieces of the smooth pp functions. We will only consider C1 pp approximation with r = 1 and k = 4. The main result in this paper is an efficient method for triangulating any finitely many arbitrarily scattered sample sites, such that these sample sites are the only vertices of the triangulation, and that for any discrete data given at these sample sites, there is a C1 piecewise quartic polynomial on this triangulation that interpolates the given data with the fifth order of approximation.

Bivariate and multivariate statistical analyses were used to predict the spatial distribution of landslides in the Cuyahoga River watershed, northeastern Ohio, U.S.A. The relationship between landslides and various instability factors contributing to their occurrence was evaluated using a Geographic Information System (GIS) based investigation. A landslide inventory map was prepared using landslide locations identified from aerial photographs, field checks, and existing literature. Instability factors such as slope angle, soil type, soil erodibility, soil liquidity index, landcover pattern, precipitation, and proximity to stream, responsible for the occurrence of landslides, were imported as raster data layers in ArcGIS, and ranked using a numerical scale corresponding to the physical conditions of the region. In order to investigate the role of each instability factor in controlling the spatial distribution of landslides, both bivariate and multivariate models were used to analyze the digital dataset. The logistic regression approach was used in the multivariate model analysis. Both models helped produce landslide susceptibility maps and the suitability of each model was evaluated by the area under the curve method, and by comparing the maps with the known landslide locations. The multivariate logistic regression model was found to be the better model in predicting landslide susceptibility of this area. The logistic regression model produced a landslide susceptibility map at a scale of 1:24,000 that classified susceptibility into four categories: low, moderate, high, and very high. The results also indicated that slope angle, proximity to stream, soil erodibility, and soil type were statistically significant in controlling the slope movement.

In this paper, we discuss the algebraic structure of bivariate C 1 cubic spline spaces over nonuniform type-2 triangulation and its subspaces with boundary conditions. The dimensions of these spaces are determined and their local support bases are constructed.

The survival of our nation, during a nuclear exchange, depends upon an effective national defense structure. The prime weapon system in this defense structure is the ballistic missile. Although many factors enter into an evaluation of the effectiveness of a ballistic missile, one of the most important measure is accuracy. Without an accurate weapon system we have no weapon system. The Department of Defense has places emphasis on using a method of accuracy evaluation called "Probably Circular Error (CEP)." Probably Circular Error is defined as "The radius of a circle, centered at the intended target, within which 50% of the missiles would be expected to impact" or "The probability is 0.50 that an individual missile will impact within a circle whose radius is equal to the CEP." The statistical techniques and assumptions used in generation a CEP value will be investigated.

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It is well-known that the basic properties of a bivariate spline space such as dimension and approximation order depend on the geometric structure of the partition. The dependence of geometric structure results in the fact that the dimension of a C1 cubic spline space over an arbitrary triangulation becomes a well-known open problem. In this paper, by employing a new group of smoothness conditions and conformality conditions, we determine the dimension of bivariate C1 cubic spline spaces over a so-called even stratified triangulation.

The present work generalizes the necessary condition of univariate symmetry of Patil et al. (2012) to the bivariate setting, develops a test of bivariate symmetry based on it, and generalizes the measure of asymmetry in Patil et al. (2014) to the bivariate setting. In doing so, as a byproduct, it pays attention to the interrelation between central symmetry and symmetry about an axis of a continuous bivariate density function.

The situation that China's pensions cannot cover the expenditure is becoming more and more serious. On the one hand, it is to increase the rate of return of enterprise annuities. On the other hand, it is to increase the participation rate of enterprise annuities. In order to improve the participation rate of enterprise annuity, it is of great significance to solve the problem of insufficient funds of enterprise annuity from the aspect of supply. Based on this, this dissertation conducts detailed research, focusing on the impact of relevant factors on the participation rate of enterprise annuity from the enterprise level and the individual level. After making a detailed overview of the situation of domestic enterprise annuity, this dissertation analyzes the influencing factors of the participation rate of enterprise annuity of China based on the latest nationally representative micro data CHIP 2018. The analysis uses the "bivariate Probit model", that is, the enterprise level and the individual level, employing 0-1 bivariate as the dependent variables and employee income, employee education, employee gender, enterprise income as independent variables. The analysis also includes other control variables. The empirical evidence found that employee income, employee education, and employee gender have impacts on employee participation in enterprise annuity; enterprise ownership, company income, and the industry in which the company is located have impacts on whether the enterprise establishes an enterprise annuity plan. In addition, there are differences in the state-owned system and private system and industry differences on the establishment of enterprise annuity plans. Finally, this dissertation gives relevant policy suggestions that may provide some theoretical guidance for increasing the participation rate of enterprise annuity.Key words: Enterprise annuity; Participation rate; Bivariate Probit model; Policy suggestions. / Business Administration/Interdisciplinary

In this thesis, we suggest a goodness-ofit test for semi-parametric copula models. We extended the pseudo in-and-out-sample (PIOS) test proposed in [17], which is based on the PIOS test in [28]. The PIOS test is constructed by comparing the pseudo "in-sample" likelihood and pseudo "out-of-sample" likelihood. Our contribution is twoold. First, we use the approximate test statistics instead of the exact test statistics to alleviate the computational burden of calculating the test statistics. Secondly, we propose a parametric bootstrap procedure to approximate the distribution of the test statistic. Unlike the nonparametric bootstrap which resamples from the original data, the parametric procedure resamples the data from the copula model under the null hypothesis. We conduct simulation studies to investigate the performance of the approximate test statistic and parametric bootstrap. The results show that the parametric bootstrap presents higher test power with a well-controlled type I error compared to the nonparametric bootstrap.