Suppose $V_i$ and $X_i$ are two continuous random variables that are jointly distributed. Let the conditional PDF and CDF of $V_i$ given $X_i=x$ be given by $f_{V|X}(V_i\mid X_i=x)$ and $F_{V|X}(V_i|X_i=x)$, respectively. For simplicity, assume $V_i$ and $X_i$ are scalar.
Consider $F_{V|X}(X_i\mid X_i=x)$. I want to know if there are standard restrictions that can be put on the conditional distribution to conclude some monotonicity property. That is, we know that since is a CDF is increasing in the argument, but also changing the argument affects the functional form. So maybe there are some conditions that can be imposed in order to imply monotonicity (weaker, of course, than independence).
If those conditions exist, I wonder if there are related restrictions that imply that that the following version of the Mills ratio: $f_{V|X}(X_i\mid X_i=x) / F_{V|X}(X_i \mid X_i=x)$ is also monotone in $X_i$. For example, if both random variables are independent and $F$ is log-concave, then the MR is decreasing in the argument. In this case, log-concavity is not enough since, as above, changing the argument changes the functional form. So there may be some properties that ensure that that effect does not dominate the other one.
Thanks a lot!