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@
