The Annals of Statistics

Asymptotic normality of the Lk-error of the Grenander estimator

Vladimir N. Kulikov and Hendrik P. Lopuhaä

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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.

Article information

Ann. Statist., Volume 33, Number 5 (2005), 2228-2255.

First available in Project Euclid: 25 November 2005

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Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

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


Kulikov, Vladimir N.; Lopuhaä, Hendrik P. Asymptotic normality of the L k -error of the Grenander estimator. Ann. Statist. 33 (2005), no. 5, 2228--2255. doi:10.1214/009053605000000462.

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