The Annals of Statistics
- Ann. Statist.
- Volume 33, Number 5 (2005), 2228-2255.
Asymptotic normality of the Lk-error of the Grenander estimator
We investigate the limit behavior of the Lk-distance between a decreasing density f and its nonparametric maximum likelihood estimator f̂n for k≥1. Due to the inconsistency of f̂n 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 1≤k<2.5, and obtain the analogous limiting result for a modification of the Lk-distance for k≥2.5. Since the L1-distance is the area between f and f̂n, which is also the area between the inverse g of f and the more tractable inverse Un of f̂n, 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 f̂n is asymptotically equivalent to the Lk-distance between Un and g.
Ann. Statist., Volume 33, Number 5 (2005), 2228-2255.
First available in Project Euclid: 25 November 2005
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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. https://projecteuclid.org/euclid.aos/1132936562