Carlo Sempi

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.

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.

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.

2-Increasing binary aggregation operators

Abstract In this work we investigate the class of binary aggregation operators (=agops) satisfying the 2-increasing property, obtaining some characterizations for agops having other special properties (e.g., quasi-arithmetic mean, Choquet-integral based, modularity) and presenting some construction methods. In particular, the notion of P -increasing function is used in order to characterize the composition of 2-increasing agops. The lattice structure (with respect to the pointwise order) of some subclasses of 2-increasing agops is presented. Finally, a method is given for constructing copulas beginning from 2-increasing and 1-Lipschitz agops.

Copulas Constructed from Horizontal Sections

In analogy with the study of copulas whose diagonal sections have been fixed, we study the set 𝒞h of copulas for which a horizontal section h has been given. We first show that this set is not empty, by explicitly writing one such copula, which we call horizontal copula. Then we find the copulas that bound both below and above the set 𝒞h. Finally, we determine the expressions for Kendalls tau and Spearmans rho for the horizontal and the bounding copulas.

Completion of probabilistic normed spaces

We prove that every probabilistic normed space, either according to the original definition given by erstnev, or according to the recent one introduced by Alsina, Schweizer and Sklar, has a completion.

Probabilistic norms and convergence of random variables

Abstract We prove that the probabilistic norms of suitable Probabilistic Normed spaces induce convergence in probability, L p convergence and almost sure convergence.