We consider kernel estimation of a smooth density $f$ at a point, but depart from the usual approach in admitting an adaptive dependence of the sharpness of the kernels on the underlying density. Proportionally varying the bandwidths like $f^{-1/2}$ at the contributing readings lowers the bias to a vanishing fraction of the usual value, and makes for performance seen in well-known estimators that force moment conditions on the kernel (and so sacrifice positivity of the curve estimate). Issues of equivariance and variance stabilitization are treated.
@article{1176345986,
author = {Abramson, Ian S.},
title = {On Bandwidth Variation in Kernel Estimates-A Square Root Law},
journal = {Ann. Statist.},
volume = {10},
number = {1},
year = {1982},
pages = { 1217-1223},
language = {en},
url = {http://dml.mathdoc.fr/item/1176345986}
}
Abramson, Ian S. On Bandwidth Variation in Kernel Estimates-A Square Root Law. Ann. Statist., Tome 10 (1982) no. 1, pp. 1217-1223. http://gdmltest.u-ga.fr/item/1176345986/