Dimitris N. Politis

The Stationary Bootstrap

Abstract This article introduces a resampling procedure called the stationary bootstrap as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on weakly dependent stationary observations. Previously, a technique based on resampling blocks of consecutive observations was introduced to construct confidence intervals for a parameter of the m-dimensional joint distribution of m consecutive observations, where m is fixed. This procedure has been generalized by constructing a “blocks of blocks” resampling scheme that yields asymptotically valid procedures even for a multivariate parameter of the whole (i.e., infinite-dimensional) joint distribution of the stationary sequence of observations. These methods share the construction of resampling blocks of observations to form a pseudo-time series, so that the statistic of interest may be recalculated based on the resampled data set. But in the context of applying this method to stationary data, it is natural...

Automatic Block-Length Selection for the Dependent Bootstrap

Abstract We review the different block bootstrap methods for time series, and present them in a unified framework. We then revisit a recent result of Lahiri [Lahiri, S. N. (1999b). Theoretical comparisons of block bootstrap methods, Ann. Statist. 27:386–404] comparing the different methods and give a corrected bound on their asymptotic relative efficiency; we also introduce a new notion of finite-sample “attainable” relative efficiency. Finally, based on the notion of spectral estimation via the flat-top lag-windows of Politis and Romano [Politis, D. N., Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Series Anal. 16:67–103], we propose practically useful estimators of the optimal block size for the aforementioned block bootstrap methods. Our estimators are characterized by the fastest possible rate of convergence which is adaptive on the strength of the correlation of the time series as measured by the correlogram.

Computer-intensive methods in statistical analysis

As far back as the late 1970s, the impact of affordable, high-speed computers on the theory and practice of modern statistics was recognized by Efron (1979, 1982). As a result, the bootstrap and other computer-intensive statistical methods (such as subsampling and the jackknife) have been developed extensively since that time and now constitute very powerful (and intuitive) tools to do statistics with. This article provides a readable, self-contained introduction to the bootstrap and jackknife methodology for statistical inference; in particular, the focus is on the derivation of confidence intervals in general situations. A guide to the available bibliography on bootstrap methods is also offered.

Bootstrap technology and applications

Bootstrap resampling methods have emerged as powerful tools for constructing inferential procedures in modern statistical data analysis. Although these methods depend on the availability of fast, inexpensive computing, they offer the potential for highly accurate methods of inference. Moreover, they can even eliminate the need to impose a convenient statistical model that does not have a strong scientific basis. In this article, we review some bootstrap methods, emphasizing applications through illustrations with some real data. Special attention is given to regression, problems with dependent data, and choosing tuning parameters for optimal performance.

Residual-Based Block Bootstrap for Unit Root Testing

A nonparametric, residual-based block bootstrap procedure is proposed in the context of testing for integrated (unit root) time series. The resampling procedure is based on weak assumptions on the dependence structure of the stationary process driving the random walk and successfully generates unit root integrated pseudo-series retaining the important characteristics of the data. It is more general than previous bootstrap approaches to the unit root problem in that it allows for a very wide class of weakly dependent processes and it is not based on any parametric assumption on the process generating the data. As a consequence the procedure can accurately capture the distribution of many unit root test statistics proposed in the literature. Large sample theory is developed and the asymptotic validity of the block bootstrap-based unit root testing is shown via a bootstrap functional limit theorem. Applications to some particular test statistics of the unit root hypothesis, i.e., least squares and Dickey-Fuller type statistics are given. The power properties of our procedure are investigated and compared to those of alternative bootstrap approaches to carry out the unit root test. Some simulations examine the finite sample performance of our procedure.

Correction to “Automatic Block-Length Selection for the Dependent Bootstrap” by D. Politis and H. White

A correction on the optimal block size algorithms of Politis and White (2004) is given following a correction of Lahiris (Lahiri 1999) theoretical results by Nordman (2008).

Full Bayesian Inference for GARCH and EGARCH Models

A full Bayesian analysis of GARCH and EGARCH models is proposed consisting of parameter estimation, model selection, and volatility prediction. The Bayesian paradigm is implemented via Markov-chain Monte Carlo methodologies. We provide implementation details and illustrations using the General Index of the Athens stock exchange.

Subsampling for heteroskedastic time series

Abstract In this article, a general theory for the construction of confidence intervals or regions in the context of heteroskedastic-dependent data is presented. The basic idea is to approximate the sampling distribution of a statistic based on the values of the statistic computed over smaller subsets of the data. This method was first proposed by Politis and Romano (1994b) for stationary observations. We extend their results to heteroskedastic observations, and prove a general asymptotic validity result under minimal conditions. In contrast, the usual bootstrap and moving blocks bootstrap are typically valid only for asymptotically linear statistics and their justification requires a case-by-case analysis. Our general asymptotic results are applied to a regression setting with dependent heteroskedastic errors.

A Warp-Speed Method for Conducting Monte Carlo Experiments Involving Bootstrap Estimators

We analyze fast procedures for conducting Monte Carlo experiments involving bootstrap estimators, providing formal results establishing the properties of these methods under general conditions.

Adaptive bandwidth choice

In this paper, we consider the problem of bandwidth choice in the parallel settings of nonparametric kernel smoothed spectral density and probability density estimation. We propose a new class of ‘plug-in’ type bandwidth estimators, and show their favorable asymptotic properties. The new estimators automatically adapt to the degree of underlying smoothness which is unknown. The main idea behind the new estimators is the use of infinite-order ‘flat-top’ kernels for estimation of the constants implicit in the formulas giving the asymptotically optimal bandwidth choices. The proposed bandwidth choice rule for ‘flat-top’ kernels has a direct analogy with the notion of thresholding in wavelets. It is shown that the use of infinite-order kernels in the pilot estimator has a twofold advantage: (a) accurate estimation of the bandwidth constants, and (b) easy determination of the required ‘pilot’ kernel bandwidth.