Skip to main content

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=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@6410@