Density estimation using spline projection kernels

Density estimation is considered using a kernel derived from an L2 projection onto certain spline spaces. The projection can be written as fp(x) = ∊ K(x, y) f(y)dy. An estimate based on asample X1,…,Xn, from f is given by p(x)=n−1∑iK(x,Xi). The pointwise asymptotic bias and variance are derived. It is found that a quadratic spline projection kernel estimator has the same asymptotic bias as a histospline does, and it outperforms the histospline in the asymptotic variance.