For the univariate current status and, more generally, the interval censoring
model, distribution theory has been developed for the maximum likelihood
estimator (MLE) and smoothed maximum likelihood estimator (SMLE) of the unknown
distribution function, see, e.g., [12], [7], [4], [5], [6], [10], [11] and [8].
For the bivariate current status and interval censoring models distribution
theory of this type is still absent and even the rate at which we can expect
reasonable estimators to converge is unknown. We define a purely discrete
plug-in estimator of the distribution function which locally converges at rate
n^{1/3} and derive its (normal) limit distribution. Unlike the MLE or SMLE,
this estimator is not a proper distribution function. Since the estimator is
purely discrete, it demonstrates that the n^{1/3} convergence rate is in
principle possible for the MLE, but whether this actually holds for the MLE is
still an open problem. If the cube root n rate holds for the MLE, this would
mean that the local 1-dimensional rate of the MLE continues to hold in
dimension 2, a (perhaps) somewhat surprising result. The simulation results do
not seem to be in contradiction with this assumption, however. We compare the
behavior of the plug-in estimator with the behavior of the MLE on a sieve and
the SMLE in a simulation study. This indicates that the plug-in estimator and
the SMLE have a smaller variance but a larger bias than the sieved MLE. The
SMLE is conjectured to have a n^{1/3}-rate of convergence if we use bandwidths
of order n^{-1/6}. We derive its (normal) limit distribution, using this
assumption. Finally, we demonstrate the behavior of the MLE and SMLE for the
bivariate interval censored data of [1], which have been discussed by many
authors, see e.g., [18], [3], [2] and [15].Comment: 18 pages, 7 figures, 4 table
