Modelling Dependence with Copulas - IFOR

3.3 Concordance
Theorem 3.4. Let X and Y be continuous random variables whose copula is C.
Then Spearman’s rho for X and Y (denoted ρ X,Y or ρ C ) is given by
ρ X,Y = ρ C = 3Q(C, Π), (3.3.8)
∫∫
= 12 uv dC(u, v) − 3, (3.3.9)
I
∫∫
2
= 12 C(u, v)du dv − 3. (3.3.10)
I 2
Note that the integral in (3.3.9) is just the expected value of the product of two
uniform (0, 1) random variables U and V whose joint distribution is the copula C.
Thus
∫∫
ρ X,Y = ρ C = 12 uv dC(u, v) − 3=12E(U, V ) − 3
I 2
=
E(UV) − 1/4
1/12
=
E(UV) − E(U)E(V )
√
Var(U)
√
Var(V )
.
If x and y are observations from random variables X and Y with distribution
functions F and G, respectively, then the “grades” (called ranks for the sample
analogue) of x and y are given by u = F (x) andv = G(y). Note the the grades are
observations from the uniform (0, 1) random variables U = F (X) andV = G(Y )
whose joint distribution function is the copula C. Hence we have (let ρ l denote the
linear correlation coefficient)
ρ X,Y = ρ l (F (X),G(Y ))
This also provides an easy way to approximatively calculate ρ X,Y
from (X, Y ) and the margins.
given a sample
Definition 10. A numeric measure κ of association between two continuous random
variables X and Y whose copula is C is a measure of concordance if it satisfies
the following properties (we use κ X,Y or κ C when convenient):
1. κ is defined for every pair X, Y of continuous random variables;
2. −1 ≤ κ X,Y ≤ 1, κ X,X =1andκ X,−X = −1;
3. κ X,Y = κ Y,X ;
4. if X and Y are independent, then κ X,Y = κ Π =0;
5. κ −X,Y = κ X,−Y = −κ X,Y ;
6. if C 1 and C 2 are copulas such that C 1 ≺ C 2 ,thenκ C1 ≤ κ C2 ;
7. If {(X n ,Y n )} is a sequence of continuous random variables with copulas
C n , and if {C n } converges pointwise to C, then lim n→∞ κ Cn = κ C .
Theorem 3.5. Let κ be a measure of concordance for continuous random variables
X and Y .
1. if Y is almost surely an increasing function of X, thenκ X,Y = κ M =1;
2. if Y is almost surely a decreasing function of X, thenκ X,Y = κ W = −1;
3. if α and β are almost surely strictly increasing functions on RanX
and RanY , respectively, then κ α(X),β(Y ) = κ X,Y .
In the next theorem we will see that Kendall’s tau and Spearman’s rho are
concordance measures according to the above definition.
Theorem 3.6. If X and Y are continuous random variables whose copula is C,
then Kendall’s tau and Spearman’s rho satisfy the properties in Definition 10 for a
measure of concordance.
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