Statistica Sinica 21 (2011), 873-899

ON THE GRENANDER ESTIMATOR AT ZERO

Fadoua Balabdaoui$^1$, Hanna Jankowski$^2$, Marios Pavlides$^3$,

Arseni Seregin$^4$ and Jon Wellner$^4$

$^1$Université Paris-Dauphine, $^2$York University, $^3$Frederick University Cyprus

and $^4$University of Washington

Abstract: We establish limit theory for the Grenander estimator of a monotone density near zero. In particular we consider the situation when the true density $f_0$ is unbounded at zero, with different rates of growth to infinity. In the course of our study we develop new switching relations using tools from convex analysis. The theory is applied to a problem involving mixtures.

Key words and phrases: Convex analysis, inconsistency, limit distribution, maximum likelihood, mixture distributions, monotone density, nonparametric estimation, Poisson process, rate of growth, switching relations.