Fabrizio Durante

Copula Theory and Its Applications

In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas. 1.1 Historical Introduction The history of copulas may be said to begin with Frechet [70]. He studied the following problem, which is stated here in dimension 2: given the distribution functions F1 and F2 of two random variables X1 and X2 defined on the same probability space (Ω ,F ,P), what can be said about the set Γ (F1,F2) of the bivariate d.f.’s whose marginals are F1 and F2? It is immediate to note that the set Γ (F1,F2), now called the Frechet class of F1 and F2, is not empty since, if X1 and X2 are independent, then the distribution function (x1,x2) → F(x1,x2) = F1(x1)F2(x2) always belongs to Γ (F1,F2). But, it was not clear which the other elements of Γ (F1,F2) were. Preliminary studies about this problem were conducted in [65, 71, 90] (see also [31, 182] for a historical overview). But, in 1959, Sklar obtained the deepest result in this respect, by introducing the notion, and the name, of a copula, and proving the theorem that now bears his name [192]. In his own words [194]: Fabrizio Durante Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Linz Austria e-mail: fabrizio.durante@jku.at Carlo Sempi Dipartimento di Matematica “Ennio De Giorgi”, Universita del Salento, Lecce, Italy e-mail: carlo.sempi@unisalento.it P. Jaworski et al. (eds.), Copula Theory and Its Applications, Lecture Notes in Statistics 198, DOI 10.1007/978-3-642-12465-5_1, c

Copula Theory: An Introduction

In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas.

Principles of copula theory

Copulas: Basic Definitions and Properties Notations Preliminaries on random variables and distribution functions Definition and first examples Characterization in terms of properties of d.f.s Continuity and absolutely continuity The derivatives of a copula The space of copulas Graphical representations Copulas and Stochastic Dependence Construction of multivariate stochastic models via copulas Sklars theorem Proofs of Sklars theorem Copulas and risk-invariant property Characterization of basic dependence structures via copulas Copulas and order statistics Copulas and Measures Copulas and d-fold stochastic measures Absolutely continuous and singular copulas Copulas with fractal support Copulas, conditional expectation, and Markov kernel Copulas and measure-preserving transformations Shuffles of a copula Sparse copulas Ordinal sums The Kendall distribution function Copulas and Approximation Uniform approximations of copulas Application to weak convergence of multivariate d.f.s Markov kernel representation and related distances Copulas and Markov operators Convergence in the sense of Markov operators The Markov Product of Copulas The Markov product Invertible and extremal elements in C2 Idempotent copulas, Markov operators, and conditional expectations The Markov product and Markov processes A generalization of the Markov product A Compendium of Families of Copulas What is a family of copulas? Frechet copulas EFGM copulas Marshall-Olkin copulas Archimedean copulas Extreme-value copulas Elliptical copulas Invariant copulas under truncation Generalizations of Copulas: Quasi-Copulas Definition and first properties Characterizations of quasi-copulas The space of quasi-copulas and its lattice structure Mass distribution associated with a quasi-copula Generalizations of Copulas: Semi-Copulas Definition and basic properties Bivariate semi-copulas, triangular norms, and fuzzy logic Relationships among capacities and semi-copulas Transforms of semi-copulas Semi-copulas and level curves Multivariate aging notions of NBU and IFR Bibliography Index

Semilinear copulas

A family of copulas, called semilinear, is constructed starting with some assumptions about the linearity of the copulas along some segments of the unit square. This family contains some other known families of copulas (e.g., Cuadras-Auge, Frechet) and has a nice statistical interpretation. Several construction methods are provided, especially concerning aggregation of semilinear copulas, and a special form of ordinal sum construction is introduced. Some results about related families of quasi-copulas and semicopulas are hence given.

COPULAS WITH GIVEN DIAGONAL SECTIONS: NOVEL CONSTRUCTIONS AND APPLICATIONS

In this paper, we present some methods for constructing copulas with a given diagonal section that are not necessarily symmetric. An interesting application for the construction of copulas with given tail dependence coefficients is, hence, provided.

A note on the convex combinations of triangular norms

We consider a property on binary operations, which we call @a-migrativity, and we use it to obtain t-norms by means of convex combinations of two t-norms, one of them being discontinuous.

On a family of copulas constructed from the diagonal section

We characterize the class of copulas that can be constructed from the diagonal section by means of the functional equation C(x,y)+|x−y|=C(x∨y,x∨y), for all (x,y) in the unit square such that C(x,y)>0. Some statistical properties of this class are given.

Rectangular Patchwork for Bivariate Copulas and Tail Dependence

We present a method for constructing bivariate copulas by changing the values that a given copula assumes on some subrectangles of the unit square. Some applications of this method are discussed, especially in relation to the construction of copulas with different tail dependencies.

Copula and semicopula transforms

We characterize the transformation, defined for every copula C, by Ch(x,y):=h[−1](C(h(x),h(y))), where x and y belong to [0,1] and h is a strictly increasing and continuous function on [0,1]. We study this transformation also in the class of quasi-copulas and semicopulas.

On a family of multivariate copulas for aggregation processes

We introduce a family of multivariate copulas - a special type of n-ary aggregation operations - depending on a univariate function. This family is used in the construction of a special aggregation operation that satisfies a Lipschitz condition. Several examples are provided and some statistical properties are studied.