Gaussian‐based kernels

Abstract

Abstract We derive a class of higher‐order kernels for estimation of densities and their derivatives, which can be viewed as an extension of the second‐order Gaussian kernel. These kernels have some attractive properties such as smoothness, manageable convolution formulae, and Fourier transforms. One important application is the higher‐order extension of exact calculations of the mean integrated squared error. The proposed kernels also have the advantage of simplifying computations of common window‐width selection algorithms such as least‐squares cross‐validation. Efficiency calculations indicate that the Gaussian‐based kernels perform almost as well as the optimal polynomial kernels when die order of the derivative being estimated is low.