Shuffles of copulas and a new measure of dependence

Ruankong, Pongpol; Santiwipanont, Tippawan; Sumetkijakan, Songkiat

doi:10.1016/j.jmaa.2012.08.061

Cited by 13 publications

(5 citation statements)

References 16 publications

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“…These dependence measures defined in terms of the copula's first partial derivatives attain the maximum value 1 at least for complete dependence copulas and the minimum value 0 when and only when the copula is Π. However, with respect to the measure of mutual complete dependence (MCD) ω, the independence copula can still be approximated by implicit dependence copulas [1], defined as copulas of random variables X and Y which are implicitly dependent in the sense that f (X) = g(Y ) a.s. for some Borel measurable functions f and g. For Rényi-type measures of dependence [13] such as ω * in [15] and ν * in [7], with respect to which all complete dependence copulas have measure 1, it can be proved that all implicit dependence copulas also have maximum measure 1. It is then evident that implicit dependence copulas plays a crucial role in understanding as well as comparing and contrasting measures of MCD and Rényi-type dependence measures.…”

Section: Introductionmentioning

confidence: 99%

Supports of Implicit Dependence Copulas

Sumetkijakan¹

2016

Preprint

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A copula of continuous random variables X and Y is called an implicit dependence copula if there exist functions α and β such that α(X) = β(Y ) almost surely, which is equivalent to C being factorizable as the * -product of a left invertible copula and a right invertible copula. Every implicit dependence copula is supported on the graph of f (x) = g(y) for some measure-preserving functions f and g but the converse is not true in general.We obtain a characterization of copulas with implicit dependence supports in terms of the non-atomicity of two newly defined associated σalgebras. As an application, we give a broad sufficient condition under which a self-similar copula has an implicit dependence support. Under certain extra conditions, we explicitly compute the left invertible and right invertible factors of the self-similar copula.

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Section: Introductionmentioning

confidence: 99%

Supports of Implicit Dependence Copulas

Sumetkijakan¹

2016

Preprint

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“…Moreover, Ruankong, Santiwipanont and Sumetkijakan (see [12]) gave the following special case frequently seen.…”

Section: The Function D Defined Bymentioning

confidence: 88%

“…Theorem 3.1 in Ruankong, Santiwipanont and Sumetkijakan [12]. Let C be a mutual complete dependence copula.…”

Section: Upper Level Functionsmentioning

confidence: 97%

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First digit distributions of partial derivatives of copulas

Junchuay

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We propose an indicator of dependence between two random variables X and Y whose distribution functions are continuous. The starting point of our investigation is the well-known fact that if C is the copula of X and Y , then the conditional expectations of Y given X and of X given Y are completely determined by the first partial derivatives of C. We then investigate whether the distributions of the first digits of @1C and @2C can give rise to a measure of dependence. Our candidates are normalized maxima of the first digit distributions of @iC. We also show that these normalized maxima attain minimum value if X and Y are independent and attain maximum value when they are completely dependent. However, both converses are not true as we are able to produce counterexamples. At last, we introduce a numerical computation of the dependence indicators. Although neither independence nor complete dependence can be characterized by the dependence indicators, we illustrate via some parametric classes of copulas that their numerical values are directly proportional to those of the dependence measures introduced by Siburg-Stoimenov and of Trutschnig.

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“…, the generalized Markov product [17] is a binary operation on the set of bivariate copulas of C and D, defined by…”

Section: Introductionmentioning

confidence: 99%

Generalized factorizability of certain implicit dependence copulas

Panyasakulwong¹

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The copula CX,Y of random variables X and Y that are uniformly distributed on [0,1] is called an implicit dependence copula if f(X)=g(Y) almost surely for some Borel functions f and g on [0,1]. Via a generalized Markov product, we give a one-to-one correspondence between the implicit dependence copulas and the parametric classes of subcopulas on a corresponding domain for the cases that f=g=Λθ, the tent function whose top is at (θ,1). We also show in the case f=g=α, a simple measure-preserving function, that implicit dependence copulas are generalized factorizable.

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Shuffles of copulas and a new measure of dependence

Cited by 13 publications

References 16 publications

Supports of Implicit Dependence Copulas

Supports of Implicit Dependence Copulas

First digit distributions of partial derivatives of copulas

Generalized factorizability of certain implicit dependence copulas

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