Skip to main content

Table 1 Bivariate Archimedean copulas, generator functions and Kendall's τ.

From: Copula based prediction models: an application to an aortic regurgitation study

Copula Generator φ (t) C(u, v) Kendall τ
Product (Independent) -ln t u·v 0
Clayton (t -θ-1/θ, θ [-1,∞)\{0} (u -θ+ v -θ-1)-1/θ θ /(θ + 2)
Gumbel (-lnt)θ, θ [1, ∞) Exp [-{(-ln u)θ+ (-ln v)θ}1/θ] (θ -1)/θ
Frank ln e t θ 1 e θ 1 , θ R MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHsislcyGGSbaBcqGGUbGBdaWcaaqaaiabdwgaLnaaCaaaleqabaGaeyOeI0IaemiDaqhcciGae8hUdehaaOGaeyOeI0IaeGymaedabaGaemyzau2aaWbaaSqabeaacqGHsislcqWF4oqCaaGccqGHsislcqaIXaqmaaGaeiilaWIae8hUdeNaeyicI4SaemOuaifaaa@4338@ 1 θ ln [ 1 + ( e u θ 1 ) ( e v θ 1 ) e θ 1 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaalmaalaaakeaajugqbiabgkHiTiabigdaXaGcbaacciqcLbuacqWF4oqCaaGagiiBaWMaeiOBa42cdaWadaGcbaqcLbuacqaIXaqmcqGHRaWklmaalaaakeaajugqbiabcIcaOiabdwgaLTWaaWbaaeqabaqcLbuacqGHsislcqWG1bqDcqWF4oqCaaGaeyOeI0IaeGymaeJaeiykaKIaeiikaGIaemyzau2cdaahaaqabeaajugqbiabgkHiTiabdAha2jab=H7aXbaacqGHsislcqaIXaqmcqGGPaqkaOqaaKqzafGaemyzau2cdaahaaqabeaajugqbiabgkHiTiab=H7aXbaacqGHsislcqaIXaqmaaaakiaawUfacaGLDbaaaaa@5502@ 1 4 θ [ 1 D 1 ( θ ) ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqaIXaqmcqGHsisldaWcaaqaaiabisda0aqaaGGaciab=H7aXbaadaWadaqaaiabigdaXiabgkHiTiabdseaenaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIae8hUdeNaeiykaKcacaGLBbGaayzxaaWaaWbaaSqabeaacqGHxiIkaaaaaa@3BD1@
  1. * D k (x) is the Debye function for any positive integer k, given by D k ( x ) = k x k 0 x t k e t 1 d t MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaajugibiabdseaeTWaaSbaaeaajugibiabdUgaRbWcbeaajugibiabcIcaOiabdIha4jabcMcaPiabg2da9SWaaSaaaOqaaKqzGeGaem4AaSgakeaajugibiabdIha4TWaaWbaaeqabaqcLbsacqWGRbWAaaaaaSWaa8qCaOqaaSWaaSaaaOqaaKqzGeGaemiDaq3cdaahaaqabeaajugibiabdUgaRbaaaOqaaKqzGeGaemyzau2cdaahaaqabeaajugibiabdsha0baacqGHsislcqaIXaqmaaaaleaajugibiabicdaWaWcbaqcLbsacqWG4baEaiabgUIiYdGaeeizaqMaemiDaqhaaa@4E68@ .