The notion of an $L_{1}$-norm density estimator process indexed by a class
of kernels is introduced. Then a functional central limit theorem and a
Glivenko--Cantelli theorem are established for this process. While
assembling the necessary machinery to prove these results, a body of
Poissonization techniques and restricted chaining methods is developed,
which is useful for studying weak convergence of general processes indexed
by a class of functions. None of the theorems imposes
any condition at all on
the underlying Lebesgue density $f$. Also, somewhat unexpectedly, the
distribution of the limiting Gaussian process does not depend on $f$.
Publié le : 2003-04-14
Classification:
Kernel density function estimator,
$L_{1}$-norm,
central limit theorem,
weak convergeance,
Poissonization,
entropy,
60F05,
60F15,
60F17,
62G07
@article{1048516534,
author = {Gin\'e, Evarist and Mason, David M. and Zaitsev, Andrei Yu.},
title = {The $\bm{L}\_\mathbf{1}$-norm density estimator process},
journal = {Ann. Probab.},
volume = {31},
number = {1},
year = {2003},
pages = { 719-768},
language = {en},
url = {http://dml.mathdoc.fr/item/1048516534}
}
Giné, Evarist; Mason, David M.; Zaitsev, Andrei Yu. The $\bm{L}_\mathbf{1}$-norm density estimator process. Ann. Probab., Tome 31 (2003) no. 1, pp. 719-768. http://gdmltest.u-ga.fr/item/1048516534/