名称

Copula分布函数

参数

Normal-Copula

C ( u , v ; ρ ) = Φ 1 ( u ) Φ 1 ( v ) 1 2 π 1 ρ 2 exp { s 2 2 ρ s t + t 2 2 ( 1 ρ 2 ) } d s d t

| ρ | < 1

t-Copula

C ( u , v ; ρ , k ) = t k 1 ( u ) t k 1 ( v ) 1 2 π 1 ρ 2 ( 1 + s 2 2 ρ s t + t 2 k ( 1 ρ 2 ) ) ( k + 2 ) / 2 d s d t

| ρ | < 1 , k > 0

Clayton

C ( u , v ; θ ) = max { ( u θ + v θ 1 ) 1 θ , 0 }

θ [ 1 , + ) \ 0

Gumbel

C ( u , v ; θ ) = exp { [ ( ln u ) θ + ( ln v ) θ ] 1 θ }

θ [ 1 , + )

Frank

C ( u , v ; θ ) = 1 θ ln ( 1 + ( e θ u 1 ) ( e θ v 1 ) e θ 1 )

θ ( , + ) \ 0