While working with rank correlation statistics of samples, specifically Spearman's rho and Kendall's tau, I came across the interesting theoretical interpretation of Kendall's tau. It can be viewed as a fundamental parameter of a bivariate distribution which essentially depends on the probability of two randomly drawn observation pairs beeing concordant.

Now, my question is the following. Is there a way to derive this probability from a bivariate distribution. How does it work? Does it need special assumptions about the form of the distribution or it is even not possible at all? And how does it look like for the theoretical value of Spearman's correlation coefficient?


The population Kendall correlation is the population probability of concordance minus population probability of discordance; $P[(X_1 − X_2) (Y_1−Y_2)>0] − P[(X_1−X_2) (Y_1−Y_2)<0]$ where $(X_1,Y_1)$ and $(X_2,Y_2)$ are independent pairs from the bivariate distribution. This corresponds directly to the sample definition (the sample proportion of concordant pairs minus the sample proportion of discordant pairs).

It is frequently used in the study of copulas; for a continuous pair of variates, $U,V$ it comes out to $\tau_{U,V} = 4 E_{U,V}[C(U,V)]-1$ where $C$ is the bivariate copula function. For example, it is used in this question Kendall's tau for Clayton Copula.

[The $U$'s and $V$'s relate to the $X$'s and $Y$'s via $U=F_X(X)$ and $V=F_Y(Y)$]

The situation for the Spearman correlation is similar though the connection to concordance probabilities is less directly obvious (nevertheless, there is such a connection).

In terms of the copula function of continuous variates the Spearman correlation can be written as $\rho_{U,V}=12E(UV)-3$.

This is perhaps easiest to motivate as follows:

Let $U=F_X(X)$ and $V=F_Y(Y)$; then $\:\,\rho = \text{Cor}(U,V) = \text{Cov}(U,V)/\sqrt{\text{Var}(U)\text{Var}(V)}=(\int_0^1\int_0^1uv\, \text{d}C(u,v)-\frac12^2)/(1/12)\\\quad=12\int_0^1\int_0^1uv\, \text{d}C(u,v)-3$

Elementary treatments of copulas sometimes explore these population correlation measures in some detail, frequently with exercises; several can be found online.

If I recall correctly some of this is discussed in Nelsen's book on copulas, but I can't check my copy right now. [Edit: see the question Population version of Kendall's tau which partly answers the first part of your question and indicates that that part at least was indeed in Nelsen. It might also be covered in Harry Joe's book.]

There's also information on these population correlation measures in Kendall's Rank Correlation Methods book.

Nelsen, R. (1999),
An Introduction to Copulas.
Lecture Notes in Statistics. Springer-Verlag, New York

Joe, H. (1997),
Multivariate Models and Dependence Concepts.
Chapman & Hall, London.


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