Durante / Sempi Principles of Copula Theory

Erscheinungsjahr 2015
ISBN: 978-1-4398-8444-7
Verlag: CRC Press
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 332 Seiten

Principles of Copula Theory explores the state of the art on copulas and provides you with the foundation to use copulas in a variety of applications. Throughout the book, historical remarks and further readings highlight active research in the field, including new results, streamlined presentations, and new proofs of old results.

After covering the essentials of copula theory, the book addresses the issue of modeling dependence among components of a random vector using copulas. It then presents copulas from the point of view of measure theory, compares methods for the approximation of copulas, and discusses the Markov product for 2-copulas. The authors also examine selected families of copulas that possess appealing features from both theoretical and applied viewpoints. The book concludes with in-depth discussions on two generalizations of copulas: quasi- and semi-copulas.

Although copulas are not the solution to all stochastic problems, they are an indispensable tool for understanding several problems about stochastic dependence. This book gives you the solid and formal mathematical background to apply copulas to a range of mathematical areas, such as probability, real analysis, measure theory, and algebraic structures.

Zielgruppe

Researchers and postgraduates in applied probability and mathematical statistics; researchers interested in applications to finance and insurance.

Autoren/Hrsg.

Durante, Fabrizio

Sempi, Carlo

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Bayesianische Inferenz

Inhaltsverzeichnis

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

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

Bibliography

Index

Über Autor(innen)

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.