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September, 1985 A Note on the $L_1$ Consistency of Variable Kernel Estimates
Luc Devroye
Ann. Statist. 13(3): 1041-1049 (September, 1985). DOI: 10.1214/aos/1176349655

Abstract

A sample $X_1, \cdots, X_n$ of i.i.d. $R^d$-valued random vectors with common density $f$ is used to construct the density estimate $f_n(x) = (1/n) \sum^n_{i = 1} H^{-d}_{ni}K((x - X_i)/H_{ni}),$ where $K$ is a given density on $R^d$, and the $H_{ni}$'s are positive functions of $n, i$ and $X_1, \cdots, X_n$ (but not of $x$). The $H_{ni}$'s can be thought of as locally adapted smoothing parameters. We give sufficient conditions for the weak convergence to 0 of $\int |f_n - f|$ for all $f$. This is illustrated for the estimate of Breiman, Meisel and Purcell (1977).

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Luc Devroye. "A Note on the $L_1$ Consistency of Variable Kernel Estimates." Ann. Statist. 13 (3) 1041 - 1049, September, 1985. https://doi.org/10.1214/aos/1176349655

Information

Published: September, 1985
First available in Project Euclid: 12 April 2007

zbMATH: 0593.62033
MathSciNet: MR803757
Digital Object Identifier: 10.1214/aos/1176349655

Subjects:
Primary: 60F15
Secondary: 62G05

Rights: Copyright © 1985 Institute of Mathematical Statistics

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Vol.13 • No. 3 • September, 1985
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