Measures of radial asymmetry for bivariate random vectors

A measure of radial asymmetry for bivariate copulas based on Sobolev norm

“…The following result shows that the index λ(C) satisfies a set of reasonable properties of a measure of radial asymmetry proposed in [3]. Proof.…”

“…which are constructed based on the L ∞ and L 2 distance between C and its survival copula C. Both measures take values in [0,1] and were first discussed by Dehgani et al [3]. If we consider the family of copulas given by (2.5), then we have Ψ ∞ (C θ/3 ) = θ and thus C 1/3 is a maximally radially asymmetric copula with respect to Ψ ∞ , while Ψ 2 (C 1/3 ) ≃ 0.46.…”

“…For instance the presence of radial asymmetry in a set of data rejects the null hypothesis of bivariate normality and other models with more flexibility should be considered. Recently several copula-based measures of radial asymmetry have been proposed and the desirable properties for such measures are addressed in [3,15]. Some tests for identifying radial symmetry of bivariate copulas are discussed in [1,10].…”

A comprehensive extension of the FGM copula

“…, 7, 2,G k (α) = G 2 k (α + ) − G 2 k (α − ), k = 1, 2, 3, and G 2 1 (x) = x − 1 − 1 2x 2 − ln x, G 2 2 (x) = x −1 − x + 2 · ln x, G 2 3 (x) = 1 2 x −2 − 2x −1 + 3 2 − ln x. For j = 2 rearrangement of integrals yields J 2 + J 3 = K 2 (α) − K 3 (α), with Dehgani et al (2013), Rosco and Joe (2013), and Genest and Nešlehová (2014).…”

A Copula-based Index to Measure Directional Reflection Asymmetry for Trivariate Copulas

“…A natural approach to measure the amount of asymmetry in a copula is based on a suitable distance between the copula and its associated survival copula. For the case d = 2, recently several copula-based measures of reflection asymmetry and statistical tests of symmetry are defined and studied in literature; see Buzebda and Cherfi (2012), Dehgani et al (2013), Genest and Neślehová (2014), and Alikhani-Vafa and Dolati (2018). Rosco and Joe (2013) studied reflection asymmetry measures using an approach based on the univariate skewness to account the direction of the asymmetry.…”