A random censored regression problem? Assume I have random variables $X$ and $Y$. $X$ is observable; $Y$ is not. We also know that, if $X \leq Y$, $Z = f(X,Y)$ where $Z$ is a third random variable and $f(\cdot)$ is a known function; otherwise $Z$ is missing. $Z$ is also observable (missing or a real number).
Assume I know the distribution of $X$ and I can estimate the distribution of $Z$ from data. I wonder if I can estimate the distribution of $Y$.
My first question is: what is this problem? It is similar to but different from a textbook censored regression problem. Is it a random censored regression problem?
My next question is, is the problem identifiable? I realize that, for any given set of random samples $X_1, X_2, X_3, ...$, there may exist multiple sequences $Y_1, Y_2, Y_3, ...$ that yield the same output $Z_1, Z_2, Z_3,...$. So I guess it will be difficult to find the best solution unless some constraints are imposed. If my guess is right, what are typical constraints people apply?
If we can appropriate apply some constraints to the problem so that we can try to find optimal parameters that minimizes some pre-defined error metric. Hopefully we can "estimate" the distribution of $Y$.
Your suggestions, books, and papers are more than welcome.
Any idea? 
 A: Consider these two simple situations:


*

*The joint distribution of $(X,Y)$ is uniform on $H^2 = [0,1] \times [0,1]$; $Z=X-Y$.

*The joint distribution of $(X, \eta)$ is uniform on $H^2$.  When $X \le \eta$, $Y=\eta$; otherwise, $Y=1$.  Again, $Z=X-Y$.
What is common to both situations are (a) the distribution of $X$ (which is uniform) and (b) the distribution of $Z$ (given that $X \le Y\ $).  Yet--obviously--the distributions of $Y$ differ radically: in the first case $Y$ is uniform, implying $\Pr[Y=1]=0$, whereas in the second case $\Pr[Y=1]=1/2$.
The point is that we are free to modify $Y$ on the set $Y\gt X$ without changing any of the information in the problem.  The information places a (crude) lower bound on the marginal CDF of $Y$, but that's all.
That answers all but the first question.  But given that the solution is so indeterminate, I doubt that problems like this (with such limited information) have been investigated, which addresses the first question.  To make progress, you have to make (parametric) assumptions about the distribution of $Y$ or the joint distribution of $(X,Y)$.
