5
$\begingroup$

I am reading about concordance and Kendall's tau. The empirical formula for Kendall's tau is given by $$ t = \frac{c-d}{c+d},$$ where $c$ and $d$ are the numbers of concordant pairs and discordant pairs in the sample.

The population version of the equation is given in Nelsen's "Introduction to Copulas" as follows: $$ \tau_{X,Y} = P[(X_1-X_2)(Y_1-Y_2)>0] - P[(X_1-X_2)(Y_1-Y_2)<0], $$ where $(X_1,Y_1)$ and $(X_2,Y_2)$ are i.i.d. random vectors each with joint distribution $H$.

This makes sense. Essentially, if we'd like to measure Kendall's tau for a bivariate distribution, we would generate two bivariate random vectors which are samples from that distribution $H$ and then calculate the probabilities as specified.

Nelsen then defines a "concordance function" $Q$. It is defined as the difference in probabilities of concordance and discordance between two vectors $(X_1,Y_1)$ and $(X_2,Y_2)$ of continuous random variables with (possibly) different joint distributions $H_1$ and $H_2$, but with common margins $F$ and $G$.

$$ Q = P[(X_1-X_2)(Y_1-Y_2)>0] - P[(X_1-X_2)(Y_1-Y_2)<0],$$ where $(X_1,Y_1)$ and $(X_2,Y_2)$ are i.i.d. random vectors with joint distribution $H_1(x,y)$ and $H_2(x,y)$.

What is the intuitive meaning of the concordance function if $H_1$ is not equal to $H_2$? Are we measuring the concordance between two different distribution functions?

$\endgroup$
1
$\begingroup$

Yes, we are measuring the difference between the probability of concordance and discordance for two observations coming from different distribution. I think $ Q(C_1,C_2) $ does not give you many informations (it turns out $ Q $ depends on the copulas, hence I used this notation) unless one of the copulas represents a "reference" copula.

For example, if you have a couple as $ (X,Y) $ with copula $ C $, you may have $ Q(C,M) $ near $1$, where $ M $ is the comonotonicity copula. This means the probability that one observation of your couple $ (X,Y) $ is concordant with an observation of the couple $ (X',Y') $ with same marginals but functionally dependent variables is much higher than the probability of discordance, hence $ Y $ tends to be higher when $ X $ is higher.

Other example: $ Q(C,\Pi)>0 $, where $ \Pi $ is the product copula. Then the probability that one observation of your couple is concordant with (lies on the 1st or 3rd quadrant with respect to) a random observation of the couple $ (X',Y') $, where $ X' $ and $ Y' $ are distributed as $X$ and $Y$ but independent, is less than the probability of discordance. So $ X $ and $ Y $ have to be somehow positively dependent. This is the idea behind Spearman's rho rank correlation coefficient.

Of course, now that you have some information about $ C $ you can use that copula as a "reference" as well.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.