Asymptotic normality of the Lk-error of the Grenander estimator

Abstract

We investigate the limit behavior of the L_k-distance between a decreasing density f and its nonparametric maximum likelihood estimator f̂_n for k≥1. Due to the inconsistency of f̂_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 L_k-distance for 1≤k<2.5, and obtain the analogous limiting result for a modification of the L_k-distance for k≥2.5. Since the L₁-distance is the area between f and f̂_n, which is also the area between the inverse g of f and the more tractable inverse U_n of f̂_n, the problem can be reduced immediately to deriving asymptotic normality of the L₁-distance between U_n and g. Although we lose this easy correspondence for k>1, we show that the L_k-distance between f and f̂_n is asymptotically equivalent to the L_k-distance between U_n and g.