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December, 1992 A Note on the Usefulness of Superkernels in Density Estimation
Luc Devroye
Ann. Statist. 20(4): 2037-2056 (December, 1992). DOI: 10.1214/aos/1176348901


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.


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Luc Devroye. "A Note on the Usefulness of Superkernels in Density Estimation." Ann. Statist. 20 (4) 2037 - 2056, December, 1992.


Published: December, 1992
First available in Project Euclid: 12 April 2007

zbMATH: 0765.62038
MathSciNet: MR1193324
Digital Object Identifier: 10.1214/aos/1176348901

Primary: 62G05

Rights: Copyright © 1992 Institute of Mathematical Statistics


Vol.20 • No. 4 • December, 1992
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