On the testability of the CAR assumption

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.