Abstract Two-dimensional distribution function g ( x, y ) defined in [0, 1] 2 is called copula, if g ( x, 1) = x and g (1, y )= y for every x, y . Similarly, s -dimensional copula is a distribution function g ( x 1 , x 2 ,..., x s ) such that every k -dimensional face function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mi>k</m:mi> </m:msub> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> g\left( {1, \ldots ,1,{x_{{i_1}}},1, \ldots ,1,{x_{{i_2}}},1, \ldots ,1,{x_{{i_k}}},1, \ldots ,1} \right) is equal to x i 1 x i 2 ... x i k for some but fixed k . In this paper we summarize and extend all known parts of copulas. In this paper we use the following abbreviations: {x} — fractional part of x ; {x} — x mod 1; [x] — integer part of x ; u.d. — uniform distribution ; d.f. — distribution function ; a.d.f. — asymptotic distribution function ; u.d.p. — uniform distribution preserving ; step d.f. — step distribution function ; a.e. — almost everywhere ; #X — cardinality of the set X .

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Copulas

N. I. Fisher · Encyclopedia of Statistical Sciences · 2005

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Copulas

Americanae (AECID Library) · 2009

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Copulas

Mary R. Hardy, David Saunders · Cambridge University Press eBooks · 2022

This well-balanced introduction to enterprise risk management integrates quantitative and qualitative approaches and motivates key mathematical and statistical methods with abundant real-world cases - both successes and failures. Worked examples and end-of-chapter exercises support readers in consolidating what they learn. The mathematical level, which is suitable for graduate and senior undergraduate students in quantitative programs, is pitched to give readers a solid understanding of the concepts and principles involved, without diving too deeply into more complex theory. To reveal the connections between different topics, and their relevance to the real world, the presentation has a coherent narrative flow, from risk governance, through risk identification, risk modelling, and risk mitigation, capped off with holistic topics - regulation, behavioural biases, and crisis management - that influence the whole structure of ERM. The result is a text and reference that is ideal for graduate and senior undergraduate students, risk managers in industry, and anyone preparing for ERM actuarial exams.

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Copulas

Peter McQuire · 2023

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Copulas

Tina Gregor · Open MIND · 2018

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Multivariate copulas, quasi-copulas and lattices

Fernández-Sánchez, Juan, Nelsen, Roger B., Úbeda-Flores, Manuel · RePEc: Research Papers in Economics

We investigate some properties of the partially ordered sets of multivariate copulas and quasi-copulas. Whereas the set of bivariate quasi-copulas is a complete lattice, which is order-isomorphic to the Dedekind-MacNeille completion of the set of bivariate copulas, we show that this is not the case in higher dimensions.

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Quasi-Copulas, Copulas and Fuzzy Implicators

Radko Mesiar, Anna Kolesárová · International Journal of Computational Intelligence Systems · 2020

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Quasi-copulas as linear combinations of copulas

Грегор Долинар, Bojan Kuzma, Nik Stopar · Fuzzy Sets and Systems · 2023

We prove that every quasi-copula can be written as a uniformly converging infinite sum of multiples of copulas. Furthermore, we characterize those quasi-copulas which can be written as a finite sum of multiples of copulas, i.e., that are a linear combination of two copulas. This generalizes a recent result of Fernández-Sánchez, Quesada-Molina, and Úbeda-Flores who considered linear combinations of discrete copulas.

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Sibuya copulas

Hofert, Marius, Vrins, Frédéric · RePEc: Research Papers in Economics

A new class of copulas referred to as “Sibuya copulas” is introduced and its properties are investigated. Members of this class are of a functional form which was first investigated in the work of M. Sibuya. The construction of Sibuya copulas is based on an increasing stochastic process whose Laplace–Stieltjes transform enters the copula as a parameter function. Sibuya copulas also allow for idiosyncratic parameter functions and are thus quite flexible to model asymmetric dependences. If the stochastic process is allowed to have jumps, Sibuya copulas may have a singular component. Depending on the choice of the process, they may be extreme-value copulas, Lévy-frailty copulas, or Marshall–Olkin copulas. Further, as a special symmetric case, one may obtain any Archimedean copula with Laplace–Stieltjes transform as generator. Besides some general properties of Sibuya copulas, several examples are given and their properties are investigated in more detail.

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Why Copulas?

Vladik Kreinovich, Hung Cuong Nguyen, Songsak Sriboonchitta et al. · scholarworks - UTEP (The University of Texas at El Paso) · 2015

A natural way to represent a 1-D probability distribution is to store its cumulative distribution function (cdf) F(x) = Prob(X ≤ x). When several random variables X1, ..., Xn are independent, the corresponding cdfs F1(x1), ..., Fn(xn) provide a complete description of their joint distribution. In practice, there is usually some dependence between the variables, so, in addition to the marginals Fi(xi), we also need to provide an additional information about the joint distribution of the given variables. It is possible to represent this joint distribution by a multi-D cdf F(x1, ..., xn) = Prob(X1 ≤ x1 & ... & Xn ≤ xn), but this will lead to duplication -- since marginals can be reconstructed from the joint cdf -- and duplication is a waste of computer space. It is therefore desirable to come up with a duplication-free representation which would still allow us to easily reconstruct F(x1, ..., xn). In this paper, we prove that the only such representation is a representation in which marginals are supplements by a copula. This result explains why copulas have been successfully used in many applications of statistics.

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DUCS copulas

Mesiar, Radko, Pekárová, Monika · Czech digital mathematics library · 2010

summary:Based on a recent representation of copulas invariant under univariate conditioning, a new class of copulas linked to a distortion of the identity function is introduced and studied.

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Copulas, Elliptical

Wiley StatsRef: Statistics Reference Online · 2014

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Gluing copulas

Siburg, Karl Friedrich, Stoimenov, Pavel A. · RePEc: Research Papers in Economics

We present a new way of constructing bivariate copulas, by recalling and gluing two (or more) copulas. Examples illustrate how this construction can be applied to build complicated copulas from simple ones.

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Factor Copulas

Martijn van der Voort · The Journal of Derivatives · 2007

Currently the industry standard model for correlated defaults is the single factor Gaussian copula, in which an issuer's default intensity comes from exposure to a common factor, which gives rise to correlation across issuers, plus an idiosyncratic component. In a rationally Bayesian world, occurrence of a default will raise the posterior expected value of the unobservable common factor, which in turn increases estimates of default correlation among the remaining credits. But this is inappropriate for defaults like those of Enron and Parmalat, that were clearly due to reasons unrelated to any common factor. In this article, van der Voort proposes modifying the Gaussian copula by adding another idiosyncratic term to deal with such cases. He shows that the model extended in this way is distinctly better able to capture the correlation skew in synthetic CDO tranche prices. <bold>TOPICS:</bold> <ext-link>CLOs, CDOs, and other structured credit</ext-link>, <ext-link>credit default swaps</ext-link>, <ext-link>statistical methods</ext-link>

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