Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask
01 Jul 2007-Journal of Hydrologic Engineering (American Society of Civil Engineers)-Vol. 12, Iss: 4, pp 347-368
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.
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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Book•
01 Jan 2010
TL;DR: The book’s methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics, to satisfactorily address extreme outcomes and the dependence of key risk drivers.
Abstract: Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book’s methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives.
1,354 citations
01 Nov 2006
TL;DR: In this paper, the authors present a critical review of blanket tests for goodness-of-fit testing of copula models and suggest new ones, and conclude with a number of practical recommendations.
Abstract: Many proposals have been made recently for goodness-of-fit testing of copula models. After reviewing them briefly, the authors concentrate on "blanket tests", i.e., those whose implementation requires neither an arbitrary categorization of the data nor any strategic choice of smoothing parameter, weight function, kernel, window, etc. The authors present a critical review of these procedures and suggest new ones. They describe and interpret the results of a large Monte Carlo experiment designed to assess the effect of the sample size and the strength of dependence on the level and power of the blanket tests for various combinations of copula models under the null hypothesis and the alternative. To circumvent problems in the determination of the limiting distribution of the test statistics under composite null hypotheses, they recommend the use of a double parametric bootstrap procedure, whose implementation is detailed. They conclude with a number of practical recommendations.
1,140 citations
TL;DR: In this paper, the authors present a critical review of the blanket test procedures and suggest new ones for goodness-of-fit testing of copula models, and describe and interpret the results of a large Monte Carlo experiment designed to assess the effect of the sample size and the strength of dependence on the level and power of blanket tests for various combinations of Copula models under the null hypothesis and the alternative.
Abstract: Many proposals have been made recently for goodness-of-fit testing of copula models. After reviewing them briefly, the authors concentrate on “blanket tests”, i.e., those whose implementation requires neither an arbitrary categorization of the data nor any strategic choice of smoothing parameter, weight function, kernel, window, etc. The authors present a critical review of these procedures and suggest new ones. They describe and interpret the results of a large Monte Carlo experiment designed to assess the effect of the sample size and the strength of dependence on the level and power of the blanket tests for various combinations of copula models under the null hypothesis and the alternative. To circumvent problems in the determination of the limiting distribution of the test statistics under composite null hypotheses, they recommend the use of a double parametric bootstrap procedure, whose implementation is detailed. They conclude with a number of practical recommendations.
995 citations
Cites methods from "Everything You Always Wanted to Kno..."
...Copula modeling has found many successful applications of late, notably in actuarial science, survival analysis and hydrology; see, e.g., Frees and Valdez (1998), Cui and Sun (2004), Genest and Favre (2007) and references therein....
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TL;DR: In this paper, Mishra et al. reviewed different methodologies used for drought modeling, which include drought forecasting, probability based modeling, spatio-temporal analysis, use of Global Climate Models (GCMs) for drought scenarios, land data assimilation systems for drought modelling, and drought planning.
706 citations
TL;DR: In this article, a multivariate, multi-index drought-modeling approach is proposed using the concept of copulas, which probabilistically combines the Standardized Precipitation Index (SPI) and the standardized Soil Moisture Index (SSI) for drought characterization.
541 citations
Cites methods from "Everything You Always Wanted to Kno..."
...The paramete r h can be estima ted from Kendall’s rank correlation s [13]:...
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...The Cramér–von Mises statistic (Sn) and Kolmogorov– Smirnov statistic (Tn) are used for goodness -of-fit tests to assess the performanc e of different copulas in modeling the dependence structure between precipita tion and soil moisture [13,15]....
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References
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Book•
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.
8,626 citations
"Everything You Always Wanted to Kno..." refers background or methods in this paper
...which are usually referred to as the Frechet‐Hoeffding bounds in the statistical literature; see, e.g., Frechet 1951 or Nelsen 1999, p. 9 . When C= W, Y is a decreasing function of X, while Y is monotone increasing in X when C= M. More generally, any copula C represents a model of dependence that lies somewhere between these two extremes, a fact that translates into the inequalities...
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...For an introduction to the theory of copulas and a large selection of related models, the reader may refer, e.g., to the monographs by Joe 1997 and Nelsen 1999, or to reviews such as Frees and Valdez 1998 and Cherubini et al. 2004, in which actuarial and financial applications are considered....
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TL;DR: Introduction.
Abstract: Introduction. Aspects of Interpretation. Technical Considerations. Statistical Analysis. Special Methods for Joint Responses. Some Examples. Strategical Aspects. More Specialized Topics. Appendices.
3,913 citations
TL;DR: This chapter reviews the main methods for generating random variables, vectors and processes in non-uniform random variate generation, and provides information on the expected time complexity of various algorithms before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.
3,304 citations
TL;DR: In this article, the authors considered the problem of estimating a U-statistic of the population characteristic of a regular functional function, where the sum ∑″ is extended over all permutations (α 1, α m ) of different integers, 1 α≤ (αi≤ n, n).
Abstract: Let X 1 …, X n be n independent random vectors, X v = , and Φ(x 1 …, x m ) a function of m(≤n) vectors . A statistic of the form , where the sum ∑″ is extended over all permutations (α1 …, α m ) of different integers, 1 α≤ (αi≤ n, is called a U-statistic. If X 1, …, X n have the same (cumulative) distribution function (d.f.) F(x), U is an unbiased estimate of the population characteristic θ(F) = f … f Φ(x 1,…, x m ) dF(x 1) … dF(x m ). θ(F) is called a regular functional of the d.f. F(x). Certain optimal properties of U-statistics as unbiased estimates of regular functionals have been established by Halmos [9] (cf. Section 4)
2,439 citations
"Everything You Always Wanted to Kno..." refers background in this paper
...and that Cn → C as n →. For more precise conditions under which this result holds, see, e.g., Hoeffding 1948 ....
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...Using Eq. 6 and the fact that under suitable regularity conditions, Cn → C as n →, one can conclude with Hoeffding 1948 that n is an asymptotically unbiased estimator of the population version of Kendall’s tau, given by...
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