Copulas have been known in the statistical literature for many years, and have become useful tools in modeling dependence structure of multivariate random variables, overcoming some of the drawbacks of the commonly-used correlation measures. Goodness-of-fit tests for copulas play a very important role in evaluating the suitability of a potential input copula model. In recent years, many approaches have been proposed for constructing goodness-of-fit tests for copula families. Among them, the so-called “blanket tests" do not require an arbitrary data categorization or any strategic choice of weight function, smoothing parameter, kernel, and so on. As preliminaries, some background and related results of copulas are firstly presented. Three goodness-of-fit test statistics belonging to the blanket test classification are then introduced. Since the asymptotic distributions of the test statistics are very complicated, parametric bootstrap procedures are employed to approximate critical values of the test statistics under the null hypotheses. To assess the performance of the three test statistics in the low dependence cases, simulation studies are carried out for three bivariate copula families, namely the Gumbel-Hougaard copula family, the Ali-Mikhail-Haq copula family, and the Farlie-Gumbel-Morgenstern copula family. Specifically the effect of low dependence on the empirical sizes and powers of the three blanket tests under various combinations of null and alternative copula families are examined. Furthermore, to check the performance of the three tests for higher dimensional copulas, the simulation studies are extended to some three-dimensional copulas. Finally the three goodness-of-fit tests are applied to two real data sets. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

Assignment (MComm)--University of Stellenbosch, 2008. / Bibliography. Also available via the Internet.

D.Comm. / Copulas provide a useful way to model different types of dependence structures explicitly. Instead of having one correlation number that encapsulates everything known about the dependence between two variables, copulas capture information on the level of dependence as well as whether the two variables exhibit other types of dependence, for example tail dependence. Tail dependence refers to the instance where the variables show higher dependence between their extreme values. A copula is defined as a multivariate distribution function with uniform marginals. A useful class of copulas is known as Archimedean copulas that are constructed from generator functions with very specific properties. The main aim of this thesis is to construct multivariate Archimedean copulas by nesting different bivariate Archimedean copulas using the vine construction approach. A characteristic of the vine construction is that not all combinations of generator functions lead to valid multivariate copulas. Established research is limited in that it presents constraints that lead to valid multivariate copulas that can be used to model positive dependence only. The research in this thesis extends the theory by deriving the necessary constraints to model negative dependence as well. Specifically, it ensures that the multivariate copulas that are constructed from bivariate copulas that capture negative dependence, will be able to capture negative dependence as well. Constraints are successfully derived for trivariate copulas. It is, however, shown that the constraints cannot easily be extended to higher-order copulas. The rules on the types of dependence structures that can be nested are also established. A number of practical applications in the financial markets where copula theory can be utilized to enhance the more established methodologies, are considered. The first application considers trading strategies based on statistical arbitrage where the information in the bivariate copula structure is utilised to identify trading opportunities in the equity market. It is shown that trading costs adversely affect the profits generated. The second application considers the impact of wrong-way risk on counterparty credit exposure. A trivariate copula is used to model the wrong-way risk. The aim of the analysis is to show how the theory developed in this thesis should be applied where negative correlation is present in a trivariate copula structure. Approaches are considered where conditional and unconditional risk driver scenarios are derived by means of the trivariate copula structure. It is argued that by not allowing for wrong-way risk, a financial institution’s credit pricing and regulatory capital calculations may be adversely affected. The final application compares the philosophy behind cointegration and copula asset allocation techniques to test which approach produces the most profitable index-tracking portfolios over time. The copula asset allocation approach performs well over time; however, it is very computationally intensive.

Thesis (Ph. D.)--University of California, San Diego, 2002. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of California, San Diego, 2007. / Title from first page of PDF file (viewed August 7, 2007). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references.

Philosophiae Doctor - PhD / The major result of this thesis is the development of a framework for the application of pair-mixtures of copulas to model asymmetric dependencies in bivariate data. The main motivation is the inadequacy of mixtures of bivariate Gaussian models which are commonly fitted to data. Mixtures of rotated single parameter Archimedean and Gaussian copulas are fitted to real data sets. The method of maximum likelihood is used for parameter estimation. Goodness-of-fit tests performed on the models giving the highest log-likelihood values show that the models fit the data well. We use mixtures of univariate Gaussian models and mixtures of regression models to investigate the existence of bimodality in the distribution of the widths of autocorrelation functions in a sample of 119 gamma-ray bursts. Contrary to previous findings, our results do not reveal any evidence of bimodality. We extend a study by Genest et al. (2012) of the power and significance levels of tests of copula symmetry, to two copula models which have not been considered previously. Our results confirm that for small sample sizes, these tests fail to maintain their 5% significance level and that the Cramer-von Mises-type statistics are the most powerful.

In this dissertation we solve the nonidentifiability problem of Archimedean copula models based on dependent censored data (see [Wang, 2012]). We give a set of identifiability conditions for a special class of bivariate frailty models. Our simulation results show that our proposed model is identifiable under our proposed conditions. We use EM algorithm to estimate unknown parameters and the proposed estimation approach can be applied to fit dependent censored data when the dependence is of research interest. The marginal survival functions can be estimated using the copula-graphic estimator (see [Zheng and Klein, 1995] and [Rivest and Wells, 2001]) or the estimator proposed by [Wang, 2014]. We also propose two model selection procedures for Archimedean copula models, one for uncensored data and the other one for right censored bivariate data. Our simulation results are similar to that of [Wang and Wells, 2000] and suggest that both procedures work quite well. The idea of our proposed model selection procedure originates from the model selection procedure for Archimedean copula models proposed by [Wang and Wells, 2000] for right censored bivariate data using the L2 norm corresponding to the Kendall distribution function. A suitable bootstrap procedure is yet to be suggested for our method. We further propose a new parameter estimator and a simple goodness-of-fit test for Archimedean copula models when the bivariate data is under fixed left truncation. Our simulation results suggest that our procedure needs to be improved so that it can be more powerful, reliable and efficient. In our strategy, to obtain estimates for the unknown parameters, we heavily exploit the concept of truncated tau (a measure of association established by [Manatunga and Oakes, 1996] for left truncated data). The idea of our goodness of fit test originates from the goodness-of-fit test for Archimedean copula models proposed by [Wang, 2010] for right censored bivariate data.

Copulas are multivariate probability distributions, as well as functions which link marginal distributions to their joint distribution. These functions have been used extensively in finance and more recently in other disciplines, for example hydrology and genetics. This study has two components, (a) the development of copula-based mathematical tools for use in all industries, and (b) the application of distorted copulas in structured finance. In the first part of this study, copulabased conditional expectation formulae are described and are applied to small data sets from medicine and hydrology. In the second part of this study we develop a method of improving the estimation of default risk in the context of collateralized debt obligations. Credit risk is a particularly important application of copulas, and given the current global financial crisis, there is great motivation to improve the way these functions are applied. We compose distortion functions with copula functions in order to obtain greater flexibility and accuracy in existing pricing algorithms. We also describe an n-dimensional dynamic copula, which takes into account temporal and spatial changes. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical sciences, 2009

Thesis (Ph. D.)--University of California, San Diego, 2005. / Vita. Includes bibliographical references (leaves 118-122).

Orientador: Verónica Andrea González-Lopez / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T10:24:16Z (GMT). No. of bitstreams: 1 Schultz_JoseAdolfodeAlmeida_M.pdf: 781903 bytes, checksum: 6f13c49a1d8a278498ea105b9b9a7a31 (MD5) Previous issue date: 2011 / Resumo: O resumo poderá ser visualizado no texto completo da tese digital / Abstract: The abstract is available with the full electronic digital document / Mestrado / Inferencia Bayesiana / Mestre em Estatística