On the simultaneous associativity ofF(x,y) andx +y -F(x,y)

TL;DR: This paper presents an introduction to inference for copula models, based on rank methods, by working out in detail a small, fictitious numerical example, the various steps involved in investigating the dependence between two random variables and in modeling it using copulas.

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Abstract: This paper presents an introduction to inference for copula models, based on rank methods. By working out in detail a small, fictitious numerical example, the writers exhibit the various steps involved in investigating the dependence between two random variables and in modeling it using copulas. Simple graphical tools and numerical techniques are presented for selecting an appropriate model, estimating its parameters, and checking its goodness-of-fit. A larger, realistic application of the methodology to hydrological data is then presented.

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1,414 citations

Cites methods from "On the simultaneous associativity o..."

...Such is the case, in particular, for several Archimedean families of copulas, e.g., those of Ali et al. 1978 , Clayton 1978 , Frank 1979 , Gumbel–Hougaard Gumbel 1960 , etc. Specifically, a copula C is said to be Archimedean if there exists a convex, decreasing function : 0,1 → 0, such that 1 =0…...
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Journal Article•DOI•

Statistical Inference Procedures for Bivariate Archimedean Copulas

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Christian Genest, Louis-Paul Rivest¹•Institutions (1)

Laval University¹

01 Sep 1993-Journal of the American Statistical Association

TL;DR: In this paper, the authors examined the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X 1, Y 1, X n, Y n ).

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Abstract: A bivariate distribution function H(x, y) with marginals F(x) and G(y) is said to be generated by an Archimedean copula if it can be expressed in the form H(x, y) = ϕ–1[ϕ{F(x)} + ϕ{G(y)}] for some convex, decreasing function ϕ defined on [0, 1] in such a way that ϕ(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thelot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription. This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X 1, Y 1), …, (X n , Y n ). The key to the estimation procedure is a one-dimensional empirical distribution function that can be constructed whether the uniform representation of X and Y is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendall's tau statistic...

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1,246 citations

Cites background or methods from "On the simultaneous associativity o..."

...For the copulas of Table 1, therefore, working with the survivor function or with the distribution itself will yield two different models-except for Frank's family, where the two are equivalent because of a peculiar symmetry condition reported by Genest (1987). Modeling of joint distributions and of survivor functions is illustrated in Section 4....
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...A general estimation procedure for Archimedean copulas should be of interest, because this class of dependence functions encompasses many well-known systems of bivariate distributions, including those of Gumbel (Gumbel 1960), Ali-Mikhail-Haq-Thelot (Ali, Mikhail, and Haq 1978; Thelot 1985), Clayton (Clayton 1978; Cook and Johnson 1981; Oakes 1982), Frank (Frank 1979; Nelsen 1986; Genest 1987), and Hougaard (1984, 1986)....
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...When 0(v) = log[ { - exp(- a) } / {- exp(-av) } ] for a real, Frank's system of bivariate distributions obtains (Frank 1979)....
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Journal Article•DOI•

A review of fuzzy set aggregation connectives

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Didier Dubois¹, Henri Prade¹•Institutions (1)

Paul Sabatier University¹

01 Jul 1985-Information Sciences

TL;DR: An extensive survey on fuzzy set-theoretic operations is provided, and the relevance of the theory of functional equations in the axiomatical construction of classes of such operations and the derivation of functional representations is emphasized.

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932 citations

Book•

Copula Modeling: An Introduction for Practitioners

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Pravin K. Trivedi¹, David M. Zimmer²•Institutions (2)

Indiana University¹, Western Kentucky University²

24 Apr 2007

TL;DR: This article explores the copula approach for econometric modeling of joint parametric distributions and demonstrates that practical implementation and estimation of copulas are relatively straightforward.

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Abstract: This article explores the copula approach for econometric modeling of joint parametric distributions. Although theoretical foundations of copulas are complex, this paper demonstrates that practical implementation and estimation are relatively straightforward. An attractive feature of parametrically specified copulas is that estimation and inference are based on standard maximum likelihood procedures, and thus copulas can be estimated using desktop econometric software. This represents a substantial advantage of copulas over recently proposed simulationbased approaches to joint modeling.

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798 citations

Book•

Risks and decisions for conservation and environmental management

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Mark A. Burgman¹•Institutions (1)

University of Melbourne¹

01 Jan 2005

TL;DR: In this paper, the authors present an overview of the risk management cycle in the context of value, history, perception, values, history and perception, and perception of uncertainty in risk management.

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Abstract: Preface Acknowledgements 1. Values, history and perception 2. Kinds of uncertainty 3. Conventions and the risk management cycle 4. Experts, stakeholders and elicitation 5. Conceptual models and hazard assessment 6. Risk ranking 7. Ecotoxicology 8. Logic trees and decisions 9. Defining and eliciting intervals 10. Monte Carlo 11. Inference, decisions, monitoring and updating 12. Decisions and risk management References Index.

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625 citations