asa.stat.comp.02

Sponsoring Section/Society: ASA-COMP

Session Slot: 8:30-10:20 Tuesday

Estimated Audience Size: 60-90

AudioVisual Request: None

Session Title: Bootstrapping Time Series

Theme Session: No

Applied Session: Yes

Session Organizer: Hesterberg, Tim MathSoft/Statistical Sciences

Address: MathSoft/Statistical Sciences 1700 Westlake Ave. N, Suite 500 Seattle, WA 98109-3044

Phone: (206)283-8802x319

Fax: (206)283-0347

Email: timh@statsci.com

Session Timing: 110 minutes total (Sorry about format):

First Speaker - 35 minutes Second Speaker - 35 minutes Third Speaker - 35 minutes Floor Discusion - 5 minutes

Session Chair: Hesterberg, Tim MathSoft/Statistical Sciences

Address: MathSoft/Statistical Sciences 1700 Westlake Ave. N, Suite 500 Seattle, WA 98109-3044

Phone: (206)283-8802x319

Fax: (206)283-0347

Email: timh@statsci.com

1. Subsampling with unknown rates of convergence

Romano, Joseph P.,   Stanford University

Address: Department of Statistics Stanford University Stanford, CA 94305-4065

Phone: 650-723-6326

Fax: 415-725-8977

Email: romano@stat.Stanford.edu

Abstract: I will give a brief overview of subsampling, as a general method for the construction of large-sample confidence regions for a general unknown parameter $\theta$ associated with the probability distribution generating a stationary sequence $X_1, \ldots, X_n$.In particular, subsampling can work when the bootstrap fails and the general theory supports this statement. The subsampling methodology hinges on approximating the large-sample distribution of a statistic $T_n = T_n(X_1, \ldots, X_n)$ that is consistent for $\theta$ at some known rate $\tau_n$.

Although subsampling has been shown to yield confidence regions for $\theta$ of asymptotically correct coverage under very weak assumptions, the application of the methodology as it has been presented so far is liminted if the rate of convergence $\tau_n$happens to be unknown or intractable in a particular setting. In this talk, I will discuss how it is possible to circumvent this limitation by (a) using the subsampling methodology to derivate a consistent estimator of the rate $\tau_n$, and (b) employing the estimated rate to construct asymptotically correct confidence regions for $\theta$ based on subsampling.

2. Matched-Block Bootstrap for Dependent Data

Kuensch, Hans-Ruedi,   Seminar fur Statistik, ETH Zurich

Address: Seminar fur Statistik ETH Zentrum CH-8092 Zurich Switzerland

Phone: +41-1-632 3416

Fax: +41-1-632 1086

Email: kuensch@stat.math.ethz.ch

Abstract: The block bootstrap for time series consists in randomly resampling blocks of consecutive values of the given data and aligning these blocks into a bootstrap sample. Hence in the bootstrap sample, different blocks are independent whereas in the original sample they are not. This impairs the performance of the blockwise bootstrap, particularaly in cases where the dependence is strong. Here we suggest a solution ot this problem by aligning with higher likelihood those blocks which match at their ends. This is achieved by resampling the blocks according to a Markov chain whose transitions depend on the data. The matching algorithms we propose take some of the dependence structure of the data into account. They are based on a kernel estimate of the conditional lag one distribution or on a fitted autoregression of small order. Numerical and theoretical analyses in the case of estimating the variance of the sample mean show that matching reduces bias and, perhaps unexpectedly, has relatively little effect on variance. Our theory extends to the case of smooth functions of a vector mean.

3. Estimation and Prediction for Time Series Using the Bootstrap

Thombs, Lori,   University of South Carolina

Address: Department of Statistics University of South Carolina Columbia, SC 29208-0001

Phone:

Fax:

Email: thombs@math.sc.edu

Abstract: We discuss two problems in time series analysis in which the bootstrap proves to be a useful tool. It is a well known result that if a process is pure white noise, then the autocorrelation estimates, scaled by the square root of the sample size, are asymptotically standard normal. We show that if the underlying process is assumed to be uncorrelated rather than independent, the asymptotic distribution is not necessarily standard normal. We present general asymptotic theory for the estimated autocorrelation function and discuss testing for lack of correlation without the further assumption of independence. Appropriate resampling methods are proposed which can be used to approximate the sampling distribution of the autocorrelation estimates under weak assumptions.

In a second application, the nonparametric bootstrap is applied to the problem of prediction in autoregresion. Standard prediction techniques for Gaussian time series utilize the result that the conditional distribution of Y[t+k], given the data, is Gaussian as well. As a nonparametric alternative, the bootstrap can be used to estimate the conditional distribution of Y[t+k]. The bootstrap replicated, being generated backward in time, all hav ethe same conditionally fixed p-values at the end of the series. Simulations are presented which illustrate the performance of the proposed bootstrap prediction technique.

List of speakers who are nonmembers: None