Increasing E(Y|X) implies increasing E(X|Y)

Question

I guessed the following statement about Conditional Expectations and tried to prove it unsuccessfully:

if $E(Y|X=x)$ is strictly increasing in $x$, then $E(X|Y=y)$ is strictly increasing in $y$.

Any hint? I also tried to find a counter-example.

I thought that this statement must be true since it seemed to me as the stochastic counterpart of the following statement: the inverse of an increasing function is also increasing. Also, it is closely related to the following: the regression coefficient of $Y$ on $X$ has the same sign that the regression coefficient of $X$ on $Y$. But I was not able to prove it yet. I also checked some parametric cases, like the joint normal distribution, where the statement is true. I also consulted some main textbooks, and google scholar but no success.

Appreciate any insight.

Answer 1

The following is a counterexample:

	Y = 0	Y = 1	Y = 100
X = 0	0%	50%	0%
X = 1	25%	0%	25%

since $E(Y|X=0) = 1$ and $E(Y|X=1) = 50$, the first condition is satisfied, but since $E(X|Y=0) = 1$ and $E(X|Y=1) = 0$, the second condition is not satisfied.