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Table 2 Algorithm steps for the Archimedean copulas.

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

Step

Clayton

Gumbel

Frank

B: θ

2τ /(1-τ)

1/(1-τ)

No closed form

C:φ(t)

(t -θ-1)/θ

(-ln t)θ

T : = ln e t θ 1 e θ 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGubavcqGG6aGocqGH9aqpcqGHsislcyGGSbaBcqGGUbGBdaWcaaqaaiabdwgaLnaaCaaaleqabaGaeyOeI0IaemiDaqhcciGae8hUdehaaOGaeyOeI0IaeGymaedabaGaemyzau2aaWbaaSqabeaacqGHsislcqWF4oqCaaGccqGHsislcqaIXaqmaaaaaa@4129@

D:φ '(t)

-θ·t -θ-1

-θ(ln t)θ -1/t

θ /(1-e tθ)

E:φ -1(t)

(1 + t)-1/θ

exp((-t)t/θ)

ln ( 1 e t + e t θ ) θ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHsisldaWcaaqaaiGbcYgaSjabc6gaUnaabmaabaGaeGymaeJaeyOeI0Iaemyzau2aaWbaaSqabeaacqGHsislcqWG0baDaaGccqGHRaWkcqWGLbqzdaahaaWcbeqaaiabgkHiTiabdsha0jabgkHiTGGaciab=H7aXbaaaOGaayjkaiaawMcaaaqaaiabeI7aXbaaaaa@40E7@

F:K c

t - ((t 1+θ- t)/θ)

t -(tln t/θ)

t ( e t θ 1 ) θ ln e t θ 1 e θ 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaajugqbiabdsha0jabgkHiTSWaaSaaaOqaaSWaaeWaaOqaaKqzafGaemyzau2cdaahaaqabeaajugqbiabdsha0HGaciab=H7aXbaacqGHsislcqaIXaqmaOGaayjkaiaawMcaaaqaaKqzafGae8hUdehaaiGbcYgaSjabc6gaUTWaaSaaaOqaaKqzafGaemyzau2cdaahaaqabeaajugqbiabgkHiTiabdsha0jab=H7aXbaacqGHsislcqaIXaqmaOqaaKqzafGaemyzau2cdaahaaqabeaajugqbiabgkHiTiab=H7aXbaacqGHsislcqaIXaqmaaaaaa@4EE7@

G:K -1 c

No closed form

No closed form

No closed form

H : ( u v ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGibascqGG6aGodaqadaqaauaabeqaceaaaeaacqWG1bqDaeaacqWG2bGDaaaacaGLOaGaayzkaaaaaa@333D@

( ( 1 + p ( t θ 1 ) ) 1 / θ ( 1 + ( 1 p ) ( t θ 1 ) ) 1 / θ ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqadaqaauaabeqaceaaaeaadaqadaqaaiabigdaXiabgUcaRiabdchaWjabgwSixlabcIcaOiabdsha0naaCaaaleqabaGaeyOeI0ccciGae8hUdehaaOGaeyOeI0IaeGymaeJaeiykaKcacaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcqaIXaqmcqGGVaWlcqWF4oqCaaaakeaadaqadaqaaiabigdaXiabgUcaRiabcIcaOiabigdaXiabgkHiTiabdchaWjabcMcaPiabgwSixlabcIcaOiabdsha0naaCaaaleqabaGaeyOeI0Iae8hUdehaaOGaeyOeI0IaeGymaeJaeiykaKcacaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcqaIXaqmcqGGVaWlcqWF4oqCaaaaaaGccaGLOaGaayzkaaaaaa@58F9@

( exp ( ( p ( ln t ) θ ) 1 / θ ) exp ( { ( 1 p ) ( ln t ) θ } 1 / θ ) ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6441@

( ln ( 1 e p T + e p T θ ) / θ ln ( 1 e ( 1 p ) T + e ( 1 p ) T θ ) / θ ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqadaqaauaabeqaceaaaeaacqGHsislcyGGSbaBcqGGUbGBdaqadaqaaiabigdaXiabgkHiTiabdwgaLnaaCaaaleqabaGaeyOeI0IaemiCaaNaemivaqfaaOGaey4kaSIaemyzau2aaWbaaSqabeaacqGHsislcqWGWbaCcqWGubavcqGHsisliiGacqWF4oqCaaaakiaawIcacaGLPaaacqGGVaWlcqWF4oqCaeaacqGHsislcyGGSbaBcqGGUbGBdaqadaqaaiabigdaXiabgkHiTiabdwgaLnaaCaaaleqabaGaeyOeI0IaeiikaGIaeGymaeJaeyOeI0IaemiCaaNaeiykaKIaemivaqfaaOGaey4kaSIaemyzau2aaWbaaSqabeaacqGHsislcqGGOaakcqaIXaqmcqGHsislcqWGWbaCcqGGPaqkcqWGubavcqGHsislcqWF4oqCaaaakiaawIcacaGLPaaacqGGVaWlcqWF4oqCaaaacaGLOaGaayzkaaaaaa@6410@