## The Annals of Statistics

### A General Resampling Scheme for Triangular Arrays of $\alpha$-Mixing Random Variables with Application to the Problem of Spectral Density Estimation

#### Abstract

In 1989 Kunsch introduced a modified bootstrap and jackknife for a statistic which is used to estimate a parameter of the $m$-dimensional joint distribution of stationary and $\alpha$-mixing observations. The modification amounts to resampling whole blocks of consecutive observations, or deleting whole blocks one at a time. Liu and Singh independently proposed (in 1988) the same technique for observations that are $m$-dependent. However, many time-series statistics, notably estimators of the spectral density function, involve parameters of the whole (infinite-dimensional) joint distribution and, hence, do not fit in this framework. In this report we generalize the "moving blocks" resampling scheme of Kunsch and Liu and Singh; a still modified version of the nonparametric bootstrap and jackknife is seen to be valid for general linear statistics that are asymptotically normal and consistent for a parameter of the whole joint distribution. We then apply this result to the problem of estimation of the spectral density.

#### Article information

Source
Ann. Statist., Volume 20, Number 4 (1992), 1985-2007.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348899

Digital Object Identifier
doi:10.1214/aos/1176348899

Mathematical Reviews number (MathSciNet)
MR1193322

Zentralblatt MATH identifier
0776.62070

JSTOR
Politis, Dimitris N.; Romano, Joseph P. A General Resampling Scheme for Triangular Arrays of $\alpha$-Mixing Random Variables with Application to the Problem of Spectral Density Estimation. Ann. Statist. 20 (1992), no. 4, 1985--2007. doi:10.1214/aos/1176348899. https://projecteuclid.org/euclid.aos/1176348899