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I am having a hard time understanding one (I assume simple) part of a proof (see the link below if you want the whole thing).

Let the two bivariate continuous random vectors: $(X_1,Y_1), (X_2,Y_2)$ be iid and $H(X,Y)$ be the joint distribution function of (,) (for index 1 as well as for index 2). $$ \Pr[X_1<X_2, Y_1<Y_2] = \int \int _{S_{X,Y}} \Pr[X_1\leq x_2, Y_1\leq y_2]dH(x,y) $$

For the left hand side, $S_{X,Y} $ refers to the support of the variable. The original post (see link below) does not include $x_2,y_2$ in the solution, it just says $x,y$ but I added them for clarity.

How does one go from the right side to the left? My attempt:

$$ =\int_{X \epsilon (-\infty,\infty)} Pr[ X_1<x_2,Y_1<Y_2 | X_2=x_2]dH(X,Y) $$ $$ =\int_{X \epsilon (-\infty,\infty)} \int_{Y \epsilon (-\infty,\infty)} Pr[ X_1<x_2,Y_1<y_2 | X_2=x_2, Y_2=y_2]dH(X,Y) $$

My question is also that I am not sure I understand that the limit is the $ -\infty,\infty$. Is it because $ X_1,X_2$ follow the same distribution? Would you please help me being very specific about the steps? Thank you.

The original post is this one. It just says

How we can prove this relation for Kendall's tau

Ahmir
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1 Answers1

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The equation $$\Pr[X_1<X_2, Y_1<Y_2] = \int \int _{S_{X,Y}} \Pr[X_1\leq x_2, Y_1\leq y_2]\,\text{d}H(x,y)$$ should indeed be $$\Pr[X_1<X_2, Y_1<Y_2] = \int _{S_{X,Y}} \Pr[X_1\leq x, Y_1\leq y]\,\text{d}H(x,y)$$ or $$\Pr[X_1<X_2, Y_1<Y_2] = \int _{S_{X,Y}} \Pr[X_1\leq x_2, Y_1\leq y_2]\,\text{d}H(x_2,y_2)$$ to be coherent. It follows from $$\Pr[X_1<X_2, Y_1<Y_2] = \mathbb{E}[\mathbb{I}_{(-\infty,X_2)}(X_1)\mathbb{I}_{(-\infty,Y_2)}(Y_1)]$$ being decomposed as \begin{align*}\Pr[X_1<X_2, Y_1<Y_2] &= \mathbb{E}^{X_2,Y_2}\{\mathbb{E}[\mathbb{I}_{(-\infty,X_2)}(X_1)\mathbb{I}_{(-\infty,Y_2)}(Y_1)]\,|\,{X_2,Y_2}\}\\ &= \mathbb{E}^{X_2,Y_2}\{\Pr[X_1<X_2, Y_1<Y_2\,|\,X_2,Y_2]\}\\ &= \int_{S_{X,Y}} \Pr[X_1<X_2, Y_1<Y_2\,|\,X_2=x_2,Y_2=y_2]\,\text{d}H(x_2,y_2)\\ &= \int _{S_{X,Y}} \Pr[X_1\leq x_2, Y_1\leq y_2]\,\text{d}H(x_2,y_2) \end{align*} since $(X_1,Y_1)$ is independent from $(X_2,Y_2)$

Xi'an
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