Problem with informative censoring 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? 
Further addition
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.
 A: This is an attempt to answer the request you made in the comments.
Independence between $T$ and $C$ vs non-informative censoring
In the following, I assume random right censoring.
Take a sample of i.i.d. survival times
$$(y_1, \delta_1), \ldots{}, (y_n, \delta_n),$$
where $y_i = \min(t_i, c_i)$ is the minimum between the survival time and the censoring time, and where $\delta_i = I(t_i \leq c_i)$ is the event indicator. So, using my notation, $T$ is the event time random variable with density $f(\cdot)$ and survival time $S(\cdot)$, while $C$ is the censoring time random variable with density $g(\cdot)$ and survival $G(\cdot)$. 
Under independence between $T$ and $C$, the likelihood function's contribution to an event time $(y_i, 1)$ is easily seen to be
$$"\Pr[T=y_i, C > y_i]" = G(y_i) f(y_i).$$
Similarly, the likelihood function's contribution to censored data $(y_i, 0)$ is
$$"\Pr[C=y_i, T > y_i]" = S(y_i) g(y_i). $$
The likelihood function for the complete data can therefore be written as
$$L = \prod_{i=1}^{n} \left[G(y_i) f(y_i)\right]^{\delta_i} \left[S(y_i) g(y_i)\right]^{1- \delta_i}.$$
Now, assume that the distribution of $C$ does not depend on the parameters of the distribution of $T$. Then the factors $G(y_i)^{\delta_i} g(y_i)^{1-\delta_i}$ are non-informative and can be factored out:
$$L \propto \prod_{i=1}^{n}  f(y_i)^{\delta_i} S(y_i)^{1- \delta_i}.$$
This is the usual likelihood when dealing with survival data. Loosely speaking, independence between $T$ and $C$ allows you to split the joint contribution of $T$ and $C$ into their marginal contributions whereas the non-informative censoring assumption allows you to get rid of $g(\cdot)$ and $G(\cdot)$.
