Estimators that satisfy certain order restrictions can arise in different ways. For example, if we want to estimate a distribution function, there are natural monotonicity constraints. But also in the context of density estimation, as well as estimation of regression curves, monotonicity constraints can arise naturally. For these situations certain isotonic estimators have been in use for a considerable time. Often these estimators can be seen as maximum likelihood estimators in a semi-parametric setting. Although conceptually these estimators have a great appeal and are easy to formulate, their distributional properties are usually of a very complicated nature.
In this talk I will focus on the nonparametric maximum likelihood estimator (NPMLE) for a decreasing density discovered by Grenander (1956). Its distributional behavior at a fixed point t > 0 (Prakasa Rao, 1969), has the striking feature that the rate of convergence is of the same order as the rate of convergence of histogram estimators, and that the asymptotic distribution is not normal. In contrast, the NPMLE is known not to be consistent at the origin. It took much longer to develop distributional theory for global measures of performance for this estimator. The first result was asymptotic normality of the L1 distance, conjectured by Groeneboom (1985) and proven more than 10 years later.
I will discuss asymptotic normality of the Lk distance for k > 1 and the behavior of the NPMLE near the origin, which yields a consistent estimator for f (0). If time permits, I will also mention some distributional results concerning the difference between the empirical distribution function and its least concave majorant.