On Bandwidth Variation in Kernel Estimates-A Square Root Law

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.