Durante / Sempi Principles of Copula Theory

Erscheinungsjahr 2015
ISBN: 978-1-4398-8444-7
Verlag: Taylor & Francis
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.