A new one parameter family of Archimedean copula and its extensions / Azam Pirmoradian - Pirmoradian, Azam (2013)
In order to characterize the dependence of extreme risk, the concept of tail dependence for bivariate distribution functions was introduced. The Gaussian copula, for example, does not have upper or lower tail dependence - it shows asymptotic independence regardless of the correlation that may exist between the variables. In other words, the extreme values in different variables occur independently even if there is a high correlation between these variables. The concept of copula aims at overcoming the tail dependence problem. The Archimedean copulas form an important family of copulas which have a simple form with properties such as associability and possess a variety of dependence structures. Specifically, the Archimedean copula for a bivariate data set can easily be constructed by a generator function. The generator uniquely determines an Archimedean copula and different choices of generator yield many families of copulas. As a consequence, many dependence properties of such copulas are relatively easy to establish because they reduce to analytical properties of the generator. Most of the Archimedean copulas with one-parameter families of generators, the Gumbel or Clayton copula for example, can explain either the upper or lower tail dependence but not both. The novelty of this thesis is to construct a new Archimedean family of copula by exploiting the properties of trigonometric functions, with an added advantage of having only one parameter. Five trigonometric copulas are constructed, namely the Cot-copula, CotII-copula, Csc-copula, CscII-copula and CscIII-copula. Our results show that these copulas have positive dependence properties which were analyzed by considering the aging properties of the respective copula. In terms of dependence properties measured vi by tail dependence and Kendall’s tau, the Cot-copula and Csc-copula are able to capture both tail dependences in symmetric and asymmetric data. Our result also shows that Cot-copula is more accurate when the lower tail dependence is heavier than the upper tail dependence, and the opposite applies to Csc-copula. Unlike the 12th family of Archimedean copula with both tail dependences, the Cot- and Csc-copula have wider dependence coverage. The advantage of Csc-copula rather than Cot-copula is its ability in capturing almost complete dependence in [0, 1]. We also extend the bivariate trigonometric copula to multivariate copula via the vine structure. For multivariate extension, the Cot-copula and Csc-copula are selected as building blocks in multivariate distribution function. The advantage of these copulas in vine structure is due to the small number of unknown parameters which reduce the estimation error especially in high dimension. Finally we demonstrate the methods developed in this study through simulation and real financial and hydrological data. In financial applications, the results show the advantage of using Cot- and Csc-copula in capturing strong tail dependences between the European market indexes. We are able to construct the multivariate dependence between the Asian markets via C-vine structure since these markets are dependent on the Singapore market index.