## A random censored regression problem?

Hello,

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?

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I'm not familiar with the term "missing." Does it mean "undefined" or "unknown"? What does it mean for $Z$ to be observable and missing at the same time? – Joel Reyes Noche Oct 12 2011 at 13:55
@joel: it is missing in the sense of en.wikipedia.org/wiki/Censoring_%28statistics%29 – psd Oct 12 2011 at 14:13
@lowerbound: Thanks. The link you provided also links to en.wikipedia.org/wiki/Missing_data. – Joel Reyes Noche Oct 12 2011 at 14:35
This question is also crossposted on stats.stackexchange.com/questions/16901/… – Joel Reyes Noche Oct 12 2011 at 14:36
In this scenario, missing means the value of $Z$ is not readily defined. I could have defined a special value, e.g. $-\infty$ though. – Haining Yu Oct 17 2011 at 17:36
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