Asymptotic normality of the Lk-error of the Grenander estimator

We investigate the limit behavior of the L-k-distance between a decreasing density f and its nonparametric maximum likelihood estimator f, for k >= 1. Due to the inconsistency of (f) over cap (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 Lk-distance for I < 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 (f) over cap (n), which is also the area between the inverse g of f and the more tractable inverse U,, of f, 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 (f) over cap (n) is asymptotically equivalent to the L-k-distance between U-n and g.