Cópula PVF: revisão de literatura e novos resultados.

Abstract

Every joint probability distribution brings with it information about the individual behavior of each variable (marginal distributions) and the dependency structure that guides the relationship between them. Statistical modeling through copula functions provides an individual analysis of these elements, enabling a detailed study of the association structures that guide a random vector. A copula of interest (and object of study of this dissertation) is the Power Variance Function (PVF) copula. It starts from the univariate three-parameter PVF distribution, derived as an extension of the positive stable distribution. The PVF copula class comprises a family of Archimedean copula that includes Clayton, Gumbel and Inverse Gaussian copulas as special cases. Through a methodological approach of literature review, it was possible to present in this work a general review of copulas, containing the main definitions, basic properties, Sklar’s theorem, Fréchet- Hoeffding bounds, dependence/association measures and parametric families, as well as identifying the main known results regarding the PVF copula, both in the bivariate and multivariate cases. In addition to the usual dependency measures (Kendall’s · and Spearman’s fl), other measures like Blomqvist’s — and Gini’s “ were investigated; in this sense, simulations of the PVF copula dependence measures with different values (i.e., different combinations of parameters) were performed. In addition, discussions were developed about the relationship of the BB9 copula with the PVF copula; specifically, we presented a demonstration that the PVF copula is a subclass of the Archimedean copula. Finally, three methods for generating data from the PVF copula and its special cases, as well as simulation and case studies, were provided.