On The Construction And Estimation Of Bivariate Copulas With Applications

The thesis mainly aims at the construction, parameter estimation and applications for bivariate copulas. The geometric weighted method is applied to construct new copulas and the interior-point penalty function algorithm is proposed to obtain maximum likelihood estimation of the parameters in geometric copulas. Those copulas including geometric copulas, conditional copulas and time-varying copulas are adopted to model the exchange rate, life expectancies and energy market data sets. We mainly discuss the estimation and application of the static copulas and conditional copulas as well as time-varying copulas.Chapter 2 presents some preliminary knowledge for copula which includes a strict definition,some properties and method on generation of random number for copulas.Chapter 3 mainly investigates the weighted geometric mean of copulas. Suitable conditions under which the weighted geometric mean of copulas is still a copula axe provided. The results further expand the scope of existing copula. We also study the properties of the weighted geometric mean of copulas. We first propose some methods to calibrate the 2-increasing property and then focus on the weighted geometric mean of the CA copula and GB copula as well as the GB copula and New copula. Weighted geometric mean of the Archimedean copulas is also investigated. The closure under weighted geometric mean is presented. Quadrant dependence and tail dependence for copulas are investigated, respectively.Chapter 4 discusses the maximum likelihood estimation of the parameters in geometric copulas and proposes an interior-point penalty function method to estimate the parameters based on their features. Finally, two real data sets including insurance data and exchange rate data are analyzed. We find that the geometric copulas outperform non-geometric copulas.Chapter 5 mainly studies the nonparametric estimation methodology for conditional copu-las. The local linear smoothing technique and Newton-Raphson method are adopted to estimate those calibration functions. Under some regularity conditions, the asymptotic normality of the estimators is obtained. Simulation study shows the efficiency of the proposed method. As an application,we analyse a life expectancies data set,the corresponding result enables us to cap-ture more precisely the conditional nonlinear correlation of expected lifetime between male and female under the level of gross domestic product(GDP).Chapter 6 mainly focuses on the dynamic specification for time-varying geometric copulas,and discusses the parameter estimation in the evolution equation and the corresponding approx-imate standard errors. In empirical analysis, the dynamic dependence structure among crude oil and natural gas prices is modeled by time-varying geometric copulas and time-invariant geomet-ric copulas as well as non-geometric cases. The corresponding results show the good performance of time-varying geometric copulas.In summary, we have concentrated on the theory and applications of the static and dynamic copulas, we also generalize some early research work. The research findings are meaningful both on theory research and practical applications.