The Annals of Statistics

Density Estimation in the $L^\infty$ Norm for Dependent Data with Applications to the Gibbs Sampler

Bin Yu

Full-text: Open access

Abstract

This paper investigates the density estimation problem in the $L^\infty$ norm for dependent data. It is shown that the iid optimal minimax rates are also optimal for smooth classes of stationary sequences satisfying certain $\beta$-mixing (or absolutely regular) conditions. Moreover, for given $\beta$-mixing coefficients, bounds on uniform convergence rates of kernel estimators are computed in terms of the mixing coefficients. The rates and the bounds obtained are not only for estimating the density but also for its derivatives. The results are then applied to give uniform convergence rates in problems associated with the Gibbs sampler.

Article information

Source
Ann. Statist., Volume 21, Number 2 (1993), 711-735.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349146

Digital Object Identifier
doi:10.1214/aos/1176349146

Mathematical Reviews number (MathSciNet)
MR1232514

Zentralblatt MATH identifier
0792.62035

JSTOR
links.jstor.org

Subjects
Primary: 62G07: Density estimation
Secondary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation

Keywords
Density estimation Gibbs sampler kernel Markov chain mixing optimal rate uniform convergence

Citation

Yu, Bin. Density Estimation in the $L^\infty$ Norm for Dependent Data with Applications to the Gibbs Sampler. Ann. Statist. 21 (1993), no. 2, 711--735. doi:10.1214/aos/1176349146. https://projecteuclid.org/euclid.aos/1176349146


Export citation