A Note on the Usefulness of Superkernels in Density Estimation

Abstract

We consider the Akaike-Parzen-Rosenblatt density estimate $f_{nh}$ based upon any superkernel $L$ (i.e., an absolutely integrable function with $\int L = 1$, whose characteristic function is 1 on $\lbrack -1, 1\rbrack)$, and compare it with a kernel estimate $g_{nh}$ based upon an arbitrary kernel $K$. We show that for a given subclass of analytic densities, $\inf_L \sup_K \lim \sup_{n\rightarrow \infty} \frac{\inf_h \mathbb{E} \int |f_{nh} - f|}{\inf_h \mathbb{E} \int |g_{nh} - f |} = 1,$ where $h > 0$ is the smoothing factor. Thus, asymptotically, the class of superkernels is as good as any other class of kernels when certain analytic densities are estimated. We also obtain exact asymptotic expressions for the expected $L_1$ error of the kernel estimate when superkernels are used.