Modelling Dependence with Copulas - IFOR

3 Dependence
3.2 Perfect Dependence
For every n-copula C we know from the Fréchet-Hoeffding inequality that
W n (u 1 ,u 2 ,... ,u n ) ≤ C(u 1 ,u 2 ,... ,u n ) ≤ M n (u 1 ,u 2 ,... ,u n ).
Furthermore, for n = 2 the upper and lower bounds are themselves copulas and we
have seen that W and M are the bivariate distributions functions of the random
vectors (U, 1 − U) and(U, U) respectively where U is uniform (0, 1). In this case we
say that W describes perfect negative dependence and M describes perfect positive
dependence.
Theorem 3.1. Let (X, Y ) have one of the copulas W or M. Then there exists two
monotonic functions α, β : R −→ R and a real-valued random variable Z so that
(X, Y )= d (α(Z),β(Z)),
with α increasing and β decreasing in the former case and both α and β increasing
in the latter case. The converse of this result is also true.
For the proof, see Embrechts, McNeil and Straumann (1999) [3].
Definition 9. If (X, Y ) has the copula M then X and Y are said to be comonotonic;
if it has the copula W they are said to be countermonotonic.
Note that if any of F 1 and F 2 (the distribution functions of X and Y respectively)
have discontinuities so that the copula is not unique, then W and M are possible
copulas.
In the case of F 1 and F 2 being continuous a stronger version of the result can
be stated:
C = W ⇔ Y = T (X) a.s., T = F2 −1 ◦ (1 − F 1 ) decreasing,
C = M ⇔ Y = T (X) a.s., T = F2 −1 ◦ F 1 increasing.
3.3 Concordance
Let (x i ,y i )and(x j ,y j ) be two observations from a random vector (X, Y ) of continuous
random variables. We say that (x i ,y i )and(x j ,y j ) are concordant if x i y j . Similarly we say that (x i ,y i )and(x j ,y j ) are
discordant if x i y j ,orifx i >x j and y i 0, and discordant
if (x i − x j )(y i − y j ) < 0.
3.3.1 Kendall’s tau
The sample version of the measure of association known as Kendall’s tau is defined
in terms of concordance as follows: Let {(x 1 ,y 1 ), (x 2 ,y 2 ),... ,(x n ,y n )} denote a
sample of n observations from a random vector (X, Y ) of continuous random variables.
Each of the ( n
2)
distinct pairs (xi ,y i )and(x j ,y j ) of observations in the sample
is either concordant or discordant. Let c and d denote the number of concordant
and discordant pairs respectively. Then Kendall’s tau for the sample is
( )
c − d
n
c + d =(c − d)/ .
2
The population version of Kendall’s tau (simply denoted Kendall’s tau) for continuous
random variables is defined similarly. Let (X 1 ,Y 1 )and(X 2 ,Y 2 ) be independent
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