We investigate the limit behavior of the Lk-distance between a decreasing density f and its nonparametric maximum likelihood estimator f̂n for k≥1. Due to the inconsistency of f̂n 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≤k<2.5, and obtain the analogous limiting result for a modification of the Lk-distance for k≥2.5. Since the L1-distance is the area between f and f̂n, which is also the area between the inverse g of f and the more tractable inverse Un of f̂n, the problem can be reduced immediately to deriving asymptotic normality of the L1-distance between Un and g. Although we lose this easy correspondence for k>1, we show that the Lk-distance between f and f̂n is asymptotically equivalent to the Lk-distance between Un and g.
"Asymptotic normality of the Lk-error of the Grenander estimator." Ann. Statist. 33 (5) 2228 - 2255, October 2005. https://doi.org/10.1214/009053605000000462