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(h²_n). 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 W_k(x)=∑^k-1₁₌₀(^k_l+1)x^lK^(l)(x)/l! and [1/∫^∞_-∞K^{(k - 1)}(x)/x d x]K^{(k - 1)}(x)/x are kernels of order k, under some mild conditions.