blomqvist beta is bounded between -1 and 1

Question

We saw during lecture the Blomqvist beta as measure of association between X and Y:

$P((X-x_0)(Y-y_0)>0)-P((X-x_0)(Y-y_0)<0)$ with $x_0 = \text{med}(X)$ and $y_0 = \text{med}(Y).$

As it is a measure of association, I assume that it is bounded between -1 and 1. Searching on the web confirmed this, but I can only find proofs that make use of copulas, with which I am not familiar.

Can anyone help me proving this without the use of copulas?

Thanks!

Answer 1

For the boundedness, I think the proof is rather more trivial than you think. Since the measure is defined as a difference between probabilities, and probabilities are bounded between $0$ and $1$, you automatically have that the difference is bounded between $-1$ and $1$.

The extreme scenarios would be if the first probability is $0$, and the second therefore $1$, thus the Blomqvist beta is $-1$. The other extreme occurs with the reverse, i.e. the first probability is $1$.