## The Annals of Statistics

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

Bin Yu

#### 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

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
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