  In recent years a popular nonparametric model for coarsened data is an
assumption on the coarsening mechanism called coarsening at random (CAR). It
has been conjectured in several papers that this assumption cannot be tested by
the data, that is, the assumption does not restrict the possible distributions
of the data. In this paper we will show that this conjecture is not always
true; an example will be current status data. We will also give conditions when
the conjecture is true, and in doing so, we will introduce a generalized
version of the CAR assumption. As an illustration, we retrieve the well-known
result that the CAR assumption cannot be tested in the case of right-censored
data.
