probability of concordance and discordance

Question

Define the probability of concordance $(\pi_c)$ and probability of discordance $(\pi_d)$. Obtain an unbiased estimate of $\tau = \pi_c-\pi_d$

I know what is concordance and discordance (usually use it to find Kendall's tau). Knowing that, I guessed that probability of concordance is $P(X_1>X_2, Y_1>Y_2)$ or something like this. But I don't really rely on guess. What I want is a article/book where it is documented or if someone gives answer of this specific question here (that will be very helpful), so that I ca answer the question clearly. Anyway, thanks for any help.

Answer 1

To quote from this textbook Chapter 5.2.3.1

More specifically, a pair of observations is concordant if the observation with the larger value of $X_1$ also has the larger value for $X_2$. If $(X_1,X_2)$ and $(X_1',X_2')$ are independent and identically distributed then they are said to be concordant if $ (X_1- X_1')(X_2-X_2')>0$ , whereas they are said to be discordant when the reverse inequality is valid. Henceforth we denote $$\text{Pr}[\text{concordance}]=\text{Pr}[(X_1 - X_1')(X_2-X_2')>0] $$ and $$ \text{Pr}[\text{discordance}]=\text{Pr}[(X_1 -X_1')(X_2-X_2')<0]. $$

In Chapter 5.2.7.1 they also discuss the issue of ties, if that is a concern for you.