Maximum smoothed likelihood estimators for the interval censoring model

We study the maximum smoothed likelihood estimator (MSLE) for interval censoring, case 2. Characterizations in terms of convex duality conditions are given and strong consistency is proved. Moreover, we show that, under smoothness conditions on the underlying distributions and using the usual bandwidth choice in density estimation, the local convergence rate is $n^{-2/5}$, in contrast with the rate $n^{-1/3}$ of the ordinary maximum likelihood estimator. In this situation the MSLE is asymptotically equivalent to the solution of a non-linear integral equation, which we solve using the implicit function theorem in Banach spaces. It is shown that, again under appropriate conditions, the MSLE has a normal limit distribution of which the expectation and variance are explicitly determined as a function of the underlying distributions. This is done by showing that the solution of the non-linear integral equation is sufficiently close to the solution of a linear integral equation, which, in turn, is sufficiently close to a "toy estimator", depending on the underlying distributions, for which we can compute the asymptotic bias and variance explicitly.