Does Tail Dependence Make A Difference In the ... - Boston College

Table 2: Bivariate Copula Functions Studied in this paper
Copula Parametric
(
Form
)
LTD UTD
Gaussian Φρ
Student t tν,ρ
(
Φ −1 (u), Φ −1 (v)
t −1
ν (u), t −1
Frank − 1 θ log (1 +
)
ν (v)
( )( )
e
−θu −1 e
−θv −1
e −θ −1
)
- -
2 − 2T1+ν( √ √
1 + ν 1−ρ
1+ρ ) 2 − 2T 1+ν( √ 1 + ν
- -
Plackett
(1 + (θ − 1)(u + v) − [(1 + (θ − 1)(u + v)) 2 − 4θuv] 1/2 ) - -
1
2 (θ − 1)−1 Clayton (u −θ + v −θ − 1) −1/θ 2 −1/θ -
Rotated-Clayton u + ( v − 1 + ((1 − u) −θ + (1 − v) −θ ) − 1) −1/θ - 2 −1/θ
Gumbel exp − [ (− log u) θ + (− log v) θ] 1 θ
- 2 − 2 1/θ
(
Rotated-Gumbel u + v − 1 + exp − [ (− log ū) θ + (− log ¯v) θ] 1 θ
2 − 2 1/θ
(
Joe-Clayton 1 − 1 − [ (1 − ū κ ) γ + (1 − ¯v κ ) −γ − 1 ] 1/κ
−1/γ)
2 −1/γ 2 −1/γ
√
1−ρ
1+ρ )
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