Asymptotically Optimum Kernels for Density Estimation at a Point

Kernel estimation of $f(0)$ is considered where f is a density in some class F of d-dimensional densities, described in terms of a Taylor series expansion. A sequence of kernels which asymptotically minimizes the maximum mean square error of estimation over F is given. The shape of the kernel is fixed, the size of the window depends on $f(0)$, and an easily computed estimate is obtained to efficiently adapt the sequence to the unknown value of $f(0)$.