I am reading "Monte Carlo Statistical Methods" by Robert and Cassella, and problem 1.3 asks
In example 1.1, the distribution of the random variable $Z=\min(X,Y)$ was of interest. Derive the distribution of $Z$ in the following case of informative censoring, where $Y\sim N(\theta,\sigma^2)$ and $X\sim N(\theta,\theta^2\sigma^2)$. Pay attention to the identifiability issues.
Now I am almost certainly missing something, but my understanding is that informative censoring happens when $X$ and $Y$ are not independent. However, just knowing that they are not independent is not enough information to get the joint distribution, but if they are independent, I do not see any identifiability issues.
Added: If $X$ and $Y$ are independent it is straightforward but tedious to write down the distribution of $Z$, the tedium exacerbated by the fact that the distribution of $X$ is a delta function when $\theta$ is $0$. For a given distribution however, we can find $\theta$ as the (obviously unique) third quartile of the distribution, and given $\theta$, $\sigma^2$ is just a scaling parameter, so there are no identifiability issues that I can see.
So, in summary, my questions are:
What precisely is the definition of informative censoring and why is the censoring in this exercise informative?
If we are meant to take $X$ and $Y$ as independent, what are the identifiability issues that need to be paid attention to?
With Ocram's explanation of informative censoring it is now clear that the identifiability issues that neede to be paid attention to were that there were not any. If the parameters for the failure and censoring distributions were separate then there would be identifiability issues as we could swap the two distributions and get the same result.
If someone more knowledgeable than I is feeling particularly Quixotic, please consider clarifying the Censoring wikipedia page.