Asymptotic normality of the L k-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 L 1-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 L 1-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 L 1-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. © Institute of Mathematical Statistics, 2005.