The Annals of Mathematical Statistics

Note on the Uniform Convergence of Density Estimates

Eugene F. Schuster

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Abstract

Let $X_1, X_2, \cdots$ be independent identically distributed random variables having a common distribution function $F$ and let $f_n(x) = (na_n)^{-1} \sum^n_{i = 1} k((x - X_i)/a_n)$ where $\{a_n\}$ is a sequence of positive numbers converging to zero and $k$ is a probability density function. If $\sum^\infty_{n = 1} \exp(- cna_n^2)$ is finite for all positive $c$ and if $k$ satisfies: (i) $k$ is continuous and of bounded variation on $(-\infty, \infty)$. (ii) $uk(u) \rightarrow 0$ as $u \rightarrow + \infty$ or $-\infty$. (iii) There exists a $\delta$ in (0, 1) such that $u(V^{-u^\delta}_{-\infty} (k) + V^\infty_{u^\delta} (k)) \rightarrow 0$ as $u \rightarrow \infty$. (iv) $\int|u| dk(u)$, the integral of $|u|$ with respect to the signed measure determined by $k$, is finite. Then the author [2] has established the following: THEOREM. A necessary and sufficient condition for $\lim_{n\rightarrow\infty} \sup_x|f_n(x) - g(x)| = 0$ with probability one for a function $g$ is that $g$ be the uniformly continuous derivative of $F$. The purpose of this note is to show that this theorem remains true if conditions (i)-(iv) on $k$ are replaced by the condition that $k$ is of bounded variation on $(-\infty, \infty)$.

Article information

Source
Ann. Math. Statist., Volume 41, Number 4 (1970), 1347-1348.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177696910

Digital Object Identifier
doi:10.1214/aoms/1177696910

Mathematical Reviews number (MathSciNet)
MR266358

Zentralblatt MATH identifier
0201.21401

JSTOR
links.jstor.org

Citation

Schuster, Eugene F. Note on the Uniform Convergence of Density Estimates. Ann. Math. Statist. 41 (1970), no. 4, 1347--1348. doi:10.1214/aoms/1177696910. https://projecteuclid.org/euclid.aoms/1177696910


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