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October 2005 Asymptotic normality of the Lk-error of the Grenander estimator
Vladimir N. Kulikov, Hendrik P. Lopuhaä
Ann. Statist. 33(5): 2228-2255 (October 2005). DOI: 10.1214/009053605000000462


We investigate the limit behavior of the Lk-distance between a decreasing density f and its nonparametric maximum likelihood estimator n for k1. Due to the inconsistency of n 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 1k<2.5, and obtain the analogous limiting result for a modification of the Lk-distance for k2.5. Since the L1-distance is the area between f and n, which is also the area between the inverse g of f and the more tractable inverse Un of n, 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 n is asymptotically equivalent to the Lk-distance between Un and g.


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Vladimir N. Kulikov. Hendrik P. Lopuhaä. "Asymptotic normality of the Lk-error of the Grenander estimator." Ann. Statist. 33 (5) 2228 - 2255, October 2005.


Published: October 2005
First available in Project Euclid: 25 November 2005

zbMATH: 1086.62063
MathSciNet: MR2211085
Digital Object Identifier: 10.1214/009053605000000462

Primary: 62E20 , 62G07
Secondary: 62G20

Keywords: Brownian motion with quadratic drift , central limit theorem , concave majorant , isotonic estimation , L_k norm , monotone density

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 5 • October 2005
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