The name is absent

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 = t_h = 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.

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