Asymptotic normality of the Lk-error of the Grenander estimator

Abstract

We investigate the limit behavior of the Lk-distance between a decreasing density f and its nonparametric maximum likelihood estimator f¿n for k¿1. Due to the inconsistency of f¿n at zero, the case k=2.5 turns out to be a kind of transition point. We extend asymptotic normality of the L1-distance to the Lk-distance for 1¿k1, we show that the Lk-distance between f and f¿n is asymptotically equivalent to the Lk-distance between Un and g.

Primary Subjects: 62E20; 62G07
Secondary Subjects: 62G20
Keywords: Brownian motion with quadratic drift; central limit theorem; concave majorant; isotonic estimation; L_k norm; monotone density