25
• Gumbel-Hougaard copula (Gumbel, 1961; Hougaard, 1986)
H(u, v; α) = exp[-{(— logu)" + (— logu)"}ly/"], a > 1. (2.22)
2.4.2 Kendall’s τ
In order to gauge the degree of association between X and Y, we use a measure known
as Kendall’s τ to measure the degree of association between variables X and Y. τ is
defined as the probability of concordance minus the probability of discordance, as
shown below:
τχ,γ = Pr[(X1 — X2)(Y1 — Y2) >0]-Pr[(Yι-X2)(Y1-Y2) < 0] (2.23)
= Pr[(X1 > X2, Y1 > Y2) or (X1 < X2, Y1 < Y2)]
-Pr[(X1 > X2, Y1 < Y2) or (X1 < X2, Y1 > Y2)]
Let X and Y be continuous random variables. Then we have,
Tχ,γ = th = Q(H, H) = 4 ʃ H(u, V, α)dH(u, v; a) - 1 (2.24)
The range of τ is (—1,1) ∖ {0}. -1 means a perfect negative correlation and 1
means a perfect positive correlation. Kendall’s τ is invariant under strictly increasing
transformations of the underlying random variables. Therefore, τ is independent from
marginal distributions, as the Equation 2.24 shows.