Principles of Copula Theory

Principles of Copula Theory

by Fabrizio Durante, Carlo Sempi



Product Details

ISBN-13: 9781439884423
Publisher: Taylor & Francis
Publication date: 07/27/2015
Pages: 332
Product dimensions: 6.12(w) x 9.25(h) x (d)

About the Author

Fabrizio Durante is a professor in the Faculty of Economics and Management at the Free University of Bozen–Bolzano. He is an associate editor of Computational Statistics & Data Analysis and Dependence Modeling. His research focuses on multivariate dependence models with copulas, reliability theory and survival analysis, and quantitative risk management. He earned a PhD in mathematics from the University of Lecce and habilitation in mathematics from the Johannes Kepler University Linz.

Carlo Sempi is a professor in the Department of Mathematics and Physics at the University of Salento. He has published nearly 100 articles in many journals. His research interests include copulas, quasi-copulas, semi-copulas, weak convergence, metric spaces, and normed spaces. He earned a PhD in applied mathematics from the University of Waterloo.

Table of Contents

Copulas: Basic Definitions and Properties
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
Sklar’s theorem
Proofs of Sklar’s 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?
Fréchet 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



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