The Annals of Statistics

An Application of the Efron-Stein Inequality in Density Estimation

Luc Devroye

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Abstract

The Efron-Stein inequality is applied to prove that the kernel density estimate $f_n$, with an arbitrary nonnegative kernel and an arbitrary smoothing factor, satisfies the inequality $\operatorname{var}(\int|f_n - f|) \leq 4/n$ for all densities $f$. Similar inequalities are obtained for other estimates.

Article information

Source
Ann. Statist., Volume 15, Number 3 (1987), 1317-1320.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176350508

Digital Object Identifier
doi:10.1214/aos/1176350508

Mathematical Reviews number (MathSciNet)
MR902261

Zentralblatt MATH identifier
0631.62039

JSTOR
links.jstor.org

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62G05: Estimation

Keywords
Efron-Stein inequality density estimation kernel estimate distribution-free confidence interval

Citation

Devroye, Luc. An Application of the Efron-Stein Inequality in Density Estimation. Ann. Statist. 15 (1987), no. 3, 1317--1320. doi:10.1214/aos/1176350508. https://projecteuclid.org/euclid.aos/1176350508


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