Bias correction and higher order kernel functions

Kernel density estimates are frequently used, based on a second order kernel. Thus, the bias inherent to the estimates has an order of O(h2n). In this note, a method of correcting the bias in the kernel density estimates is provided, which reduces the bias to a smaller order. Effectively, this method produces a higher order kernel based on a second order kernel. For a kernel function K, the functions Wk(x)=[summation operator]k-11=0(kl+1)xlK(l)(x)/l! and [1/[integral operator][infinity]-[infinity]K(k - 1)(x)/x d x]K(k - 1)(x)/x are kernels of order k, under some mild conditions.Bias correction higher order kernel kernel density estimate nonparametrics