I'm currently studying for my statistics exam by doing exercises from the book Statistics and Data Analysis for Financial Engineering with R examples.
I'm struggling with this exercise from chapter 8:
Kendall’s tau rank correlation between $X$ and $Y$ is $0.55$. Both $X$ and $Y$ are positive. What is Kendall’s tau between $X$ and $1/Y$ ? What is Kendall’s tau between $1/X$ and $1/Y$ ?
All I know is that Kendall’s tau is the probability of a concordant pair minus the probability of a discordant pair. But I don't know how to apply it here.
Any insight is appreciated.
Consider two independent pairs of random variables $(X_1,Y_1)$ and $(X_2,Y_2)$ drawn from the bivariate distribution of $(X,Y)$.
Kendall's tau between $X$ and $Y$ can be written as
$$\tau(X,Y)=E\left[\operatorname{sgn}(X_1-X_2)\operatorname{sgn}(Y_1-Y_2)\right]\,,$$
where $\operatorname{sgn}(\cdot)$ is the sign function.
By definition, observe that $\operatorname{sgn}\left(\frac1{X_1}-\frac1{X_2}\right)=-\operatorname{sgn}(X_1-X_2)$.
Based on this, you should be able to relate both $\tau\left(X,\frac1Y\right)$ and $\tau\left(\frac1X,\frac1Y\right)$ with $\tau(X,Y)$.