The Annals of Statistics

The $L_1$ Convergence of Kernel Density Estimates

L. P. Devroye and T. J. Wagner

Full-text: Open access

Abstract

Let $X_1, \cdots, X_n$ be a sequence of independent random vectors taking values in $\mathbf{\mathbb{R}}^d$ with a common probability density $f$. If $f_n(x) = (1/h) h^{-d}_n\sum^n_{i = 1}K((x - X_i)/h_n)$ is the kernel estimate of $f$ from $X_1, \cdots, X_n$ then conditions on $K$ and $\{h_n\}$ are given which insure that $\int|f_n(x) - f(x)|dx \rightarrow_n 0$ in probability or with probability one. No continuity conditions are imposed on $f$.

Article information

Source
Ann. Statist., Volume 7, Number 5 (1979), 1136-1139.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176344796

Digital Object Identifier
doi:10.1214/aos/1176344796

Mathematical Reviews number (MathSciNet)
MR536515

Zentralblatt MATH identifier
0423.62031

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 62G05: Estimation

Keywords
Density estimation integral convergence kernel estimates

Citation

Devroye, L. P.; Wagner, T. J. The $L_1$ Convergence of Kernel Density Estimates. Ann. Statist. 7 (1979), no. 5, 1136--1139. doi:10.1214/aos/1176344796. https://projecteuclid.org/euclid.aos/1176344796


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