Asymptotic normality of the Lk-error of the Grenander estimator

• 成果类型: Article
• 署名作者: Kulikov, VN; Lopuhaä, HP
• 署名单位: Delft University of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053605000000462
• 发表日期: 2005
• 页码: 2228-2255
• 关键词: density

摘要

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.