Bivariate Copulas, Transformations and Measures of Concordance

This book presents the theory of bivariate copulas and shows their potential for quantifying stochastic dependence between two random variables through measures of concordance. It emphasizes the importance of copula measures, patchwork copulas and transformations of copulas. Opening with a prologue leading to the famous theorem of Sklar, which can be considered the backbone of copula theory, the book is organized into three parts. The first part presents the foundations of bivariate copulas and copula measures, as well as the construction of patchwork copulas. The second part is devoted to transformations and measures of concordance. It introduces a group of transformations of copulas, studies invariance properties and the integral of a copula with respect to a copula measure, and then develops the theory of measures of concordance, including their comparison and estimation. The third part covers a range of special topics, including Archimedean and max-stable copulas, diagonals, distortions and shuffles of copulas, measures of dependence and asymmetry, the Markov product of copulas and measures of functional dependence, as well as order statistics and quasicopulas. The text concludes with a glance at multivariate copulas. Including problems for self-study, the book provides a solid basis for graduate students and researchers new to the field who are interested in copula theory.