Book•
An Introduction to Copulas
01 Jan 1999-
TL;DR: This book discusses the fundamental properties of copulas and some of their primary applications, which include the study of dependence and measures of association, and the construction of families of bivariate distributions.
Abstract: The study of copulas and their role in statistics is a new but vigorously growing field. In this book the student or practitioner of statistics and probability will find discussions of the fundamental properties of copulas and some of their primary applications. The applications include the study of dependence and measures of association, and the construction of families of bivariate distributions. This book is suitable as a text or for self-study.
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TL;DR: The Gaussian mixture representation of a multivariate t distribution is used as a starting point to construct two new copulas, the skewed t copula and the grouped tCopula, which allow more heterogeneity in the modelling of dependent observations.
Abstract: Summary The t copula and its properties are described with a focus on issues related to the dependence of extreme values. The Gaussian mixture representation of a multivariate t distribution is used as a starting point to construct two new copulas, the skewed t copula and the grouped t copula, which allow more heterogeneity in the modelling of dependent observations. Extreme value considerations are used to derive two further new copulas: the t extreme value copula is the limiting copula of componentwise maxima of t distributed random vectors; the t lower tail copula is the limiting copula of bivariate observations from a t distribution that are conditioned to lie below some joint threshold that is progressively lowered. Both these copulas may be approximated for practical purposes by simpler, better-known copulas, these being the Gumbel and Clayton copulas respectively.
952 citations
Book•
11 Aug 2003TL;DR: The Statistical Size Distribution in Economics and Actuarial Sciences (SDFIS) as discussed by the authors is a collection of parametric models that deal with income, wealth, and related notions.
Abstract: A comprehensive account of economic size distributions around the world and throughout the years In the course of the past 100 years, economists and applied statisticians have developed a remarkably diverse variety of income distribution models, yet no single resource convincingly accounts for all of these models, analyzing their strengths and weaknesses, similarities and differences. Statistical Size Distributions in Economics and Actuarial Sciences is the first collection to systematically investigate a wide variety of parametric models that deal with income, wealth, and related notions. Christian Kleiber and Samuel Kotz survey, compliment, compare, and unify all of the disparate models of income distribution, highlighting at times a lack of coordination between them that can result in unnecessary duplication. Considering models from eight languages and all continents, the authors discuss the social and economic implications of each as well as distributions of size of loss in actuarial applications. Specific models covered include: Pareto distributions Lognormal distributions Gamma-type size distributions Beta-type size distributions Miscellaneous size distributions Three appendices provide brief biographies of some of the leading players along with the basic properties of each of the distributions. Actuaries, economists, market researchers, social scientists, and physicists interested in econophysics will find Statistical Size Distributions in Economics and Actuarial Sciences to be a truly one-of-a-kind addition to the professional literature.
882 citations
Cites background from "An Introduction to Copulas"
...As is well known from the theory of copulas (e.g., Nelsen, 1998), a multivariate survival function can be decomposed in the form F(x1, . . . , xk) ¼ G[ F1(x1), . . . , Fk(xk)], where the Fi(xi), i ¼ 1, . . . , xk , are the marginal survival functions and G is the copula (a c.d.f. on [0, 1]k that…...
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TL;DR: A general formula for the density of a vine dependent distribution is derived, which generalizes the well-known density formula for belief nets based on the decomposition of belief nets into cliques and allows a simple proof of the Information Decomposition Theorem for a regular vine.
Abstract: A vine is a new graphical model for dependent random variables Vines generalize the Markov trees often used in modeling multivariate distributions They differ from Markov trees and Bayesian belief nets in that the concept of conditional independence is weakened to allow for various forms of conditional dependence A general formula for the density of a vine dependent distribution is derived This generalizes the well-known density formula for belief nets based on the decomposition of belief nets into cliques Furthermore, the formula allows a simple proof of the Information Decomposition Theorem for a regular vine The problem of (conditional) sampling is discussed, and Gibbs sampling is proposed to carry out sampling from conditional vine dependent distributions The so-called ‘canonical vines’ built on highest degree trees offer the most efficient structure for Gibbs sampling
836 citations
Cites methods from "An Introduction to Copulas"
...More information about copulae can be found in [17]....
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TL;DR: In this paper, an overview of the principal issues of extremal dependence is provided through a unified approach which encompasses both the limiting and independent cases of extreme dependence, and diagnostic measures for dependence are also developed.
Abstract: Quantifying dependence is a central theme in probabilistic and statistical methods for multivariate extreme values. Two situations are possible: one where, in a limiting sense, the extremes are dependent; the other where, in the same sense, the extremes are independent. This paper comprises an overview of the principal issues through a unified approach which encompasses both these situations. Novel diagnostic measures for dependence are also developed which provide complementary information about different aspects of extremal dependence. The paper is written in an elementary style, with the methodology illustrated by application to theoretical examples and typical data-sets. These data-sets and the S-plus functions used for the analyses are available online.
815 citations
TL;DR: In this article, the authors model dependence with switching-parameter copulas to study financial contagion, using daily returns from five East Asian stock indices during the Asian crisis and from four Latin American stock index during the Mexican crisis, finding evidence of changing dependence during periods of turmoil.
799 citations