We consider an arbitrary sequence of kernel density estimates $f_n$ with kernels $K_n$ possibly depending upon $n$. Under a mild restriction on the sequence $K_n$, we obtain inequalities of the type $E\big(\int|f_n - f|\big) \geq (1 + o(1))\Psi(n, f),$ where $f$ is the density being estimated and $\Psi(n, f)$ is a function of $n$ and $f$ only. The function $\psi$ can be considered as an indicator of the difficulty of estimating $f$ with any kernel estimate.
Publié le : 1988-09-14
Classification:
Density estimation,
$L_1$ error,
inequalities,
characteristic function,
kernel estimate,
performance bounds,
60E15,
62G05
@article{1176350953,
author = {Devroye, Luc},
title = {Asymptotic Performance Bounds for the Kernel Estimate},
journal = {Ann. Statist.},
volume = {16},
number = {1},
year = {1988},
pages = { 1162-1179},
language = {en},
url = {http://dml.mathdoc.fr/item/1176350953}
}
Devroye, Luc. Asymptotic Performance Bounds for the Kernel Estimate. Ann. Statist., Tome 16 (1988) no. 1, pp. 1162-1179. http://gdmltest.u-ga.fr/item/1176350953/