Abstract: The first part of this paper gives some general consistency theorems for the maximum product of spacings (MPS) method, an estimation method related to maximum likelihood. The second part deals with nonparametric estimation of a concave (convex) distribution and more generally a unimodal distribution, without smoothness assumptions on the densities. The MPS estimator for a distribution function with a monotone density is shown to have a simple explicit representation analogous to the Grenander estimator, and is asymptotically minimax with respect to Kolmogorov-Smirnov type loss. A simple consistent MPS estimator for a unimodal distribution is also discussed.
Key words and phrases: Grenander estimator, nonparametric MPS estimator, spacings, total variation distance, unimodal density.