Asymptotic Normality of the L K - 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 es-timatorˆfn for k ≥ 1. Due to the inconsistency ofˆfn at zero, the case k = 2.5 turns out to be a kind of transition point. We extend asymp-totic 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 andˆfn, which is also the area between the inverse g of f and the more tractable inverse Un ofˆfn, 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 L k-distance between f andˆfn is asymptotically equivalent to the L k-distance between Un and g. 1. Introduction. Let f be a nonincreasing density with compact support. Without loss of generality, assume this to be the interval [0, 1]. The nonparametric maximum likelihood estimatorˆf n of f was discovered by Grenander [2]. It is defined as the left derivative of the least concave ma-jorant (LCM) of the empirical distribution function F n constructed from a sample X 1 ,. .. , X n from f. Prakasa Rao [11] obtained the earliest result on the asymptotic pointwise behavior of the Grenander estimator. One immediately striking feature of this result is that the rate of convergence is of the same order as the rate of convergence of histogram estimators, and that the asymptotic distribution is not normal. It took much longer to develop distri-butional theory for global measures of performance for this estimator. The first distributional result for a global measure of deviation was the convergence to a normal distribution of the L 1-error mentioned in [3] (see [4] for a