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.
Publié le : 2005-10-14
Classification:
Brownian motion with quadratic drift,
central limit theorem,
concave majorant,
isotonic estimation,
L_k norm,
monotone density,
62E20,
62G07,
62G20
@article{1132936562,
author = {Kulikov, Vladimir N. and Lopuha\"a, Hendrik P.},
title = {Asymptotic normality of the L<sub>
k
</sub>-error of the Grenander estimator},
journal = {Ann. Statist.},
volume = {33},
number = {1},
year = {2005},
pages = { 2228-2255},
language = {en},
url = {http://dml.mathdoc.fr/item/1132936562}
}
Kulikov, Vladimir N.; Lopuhaä, Hendrik P. Asymptotic normality of the L
k
-error of the Grenander estimator. Ann. Statist., Tome 33 (2005) no. 1, pp. 2228-2255. http://gdmltest.u-ga.fr/item/1132936562/