Sufficient m-out-of-n (m/n) bootstrap

Traditional resampling methods for estimating sampling distributions sometimes fail, and alternative approaches are then needed. For example, if the classical central limit theorem does not hold and the naïve bootstrap fails, the m/n bootstrap, based on smaller-sized resamples, may be used as an alternative. An alternative to the naïve bootstrap, the sufficient bootstrap, which uses only the distinct observations in a bootstrap sample, is another recently proposed bootstrap approach that has been suggested to reduce the computational burden associated with bootstrapping. It works as long as naïve bootstrap does. However, if the naïve bootstrap fails, so will the sufficient bootstrap. In this paper, we propose combining the sufficient bootstrap with the m/n bootstrap in order to both regain consistent estimation of sampling distributions and to reduce the computational burden of the bootstrap. We obtain necessary and sufficient conditions for asymptotic normality of the proposed method, and propose new values for the resample size m. We compare the proposed method with the naïve bootstrap, the sufficient bootstrap, and the m/n bootstrap by simulation.     

A Monte Carlo comparison of three consistent bootstrap procedures

Since bootstrap samples are simple random samples with replacement from the original sample, the information content of some bootstrap samples can be very low. To avoid this fact, several variants of the classical bootstrap have been proposed. In this paper, we consider two of them: the sequential or Poisson bootstrap and the reduced bootstrap. Both of these, like the ordinary bootstrap, can yield second-order accurate distribution estimators, that is, the three bootstrap procedures are asymptotically equivalent. The question that naturally arises is which of them should be used in a practical situation, in other words, which of them should be used for finite sample sizes. To try to answer this question, we have carried out a simulation study. Although no method was found to exhibit best performance in all the considered situations, some recommendations are given.     

Bootstrapping stochastic regression models under homoskedasticity: wild bootstrap vs. pairs bootstrap

We investigate by simulation how the wild bootstrap and pairs bootstrap perform in t and F tests of regression parameters in the stochastic regression model, where explanatory variables are stochastic and not given and there exists no heteroskedasticity. The wild bootstrap procedure due to Davidson and Flachaire [The wild bootstrap, tamed at last, Working paper, IER#1000, Queen's University, 2001] with restricted residuals works best but its dominance is not strong compared to the result of Flachaire [Bootstrapping heteroskedastic regression models: wild bootstrap vs. pairs bootstrap, Comput. Statist. Data Anal. 49 (2005), pp. 361–376] in the fixed regression model where explanatory variables are fixed and there exists heteroskedasticity.     

Asymptotic properties of the bootstrap unit root test statistic under possibly infinite variance

ABSTRACT

In this article, the unit root test for the AR(1) model is discussed, under the condition that the innovations of the model are in the domain of attraction of the normal law with possibly infinite variances. By using residual bootstrap with sample size m < n (n being the size of the original sample), we bootstrap the least-squares estimator of the autoregressive parameter. Under some mild assumptions, we prove that the null distribution of the unit root test statistic based on the least-square estimator of the autoregressive parameter can be approximated by using residual bootstrap.     

Mathematical proof of the third order accuracy of the speedy double bootstrap method

Abstract

In our previous research, we proposed a speedy double bootstrap method for assessing the reliability of statistical models with maximum log-likelihood criterion. It can provide 3rd order accurate probabilities. In this study, our focus switches to the mathematical proof. We propose an alternative proof of the third order accuracy in the context of the multivariate normal model. Our proof is based on tube formula differential geometric methodology and an Taylor series approach to the asymptotic analysis of the bootstrap method.     

Non-parametric bootstrap recycling

The double bootstrap provides diagnostics for bootstrap calculations and, if need be, appropriate adjustments. The amount of computation involved is usually considerable, and recycling provides a less computer intensive alternative. Recycling consists of using repeatedly the same samples drawn from a recycling distribution G for estimation under each first-level bootstrap distribution, rather than independently repeating the simulation and estimation steps for each of these.Recycling is successful in parametric applications of the bootstrap, as demonstrated by M.A. Newton and C.J. Geyer (J. Amer. Statist. Assoc. 89: 905–912, 1994). We show that it is bound to fail in non-parametric bootstrap applications, and suggest a modification that makes the method work. The modification consists of smoothing the first-level bootstrap distributions, with the desired consequence that this removes the zero probabilities in the multinomial distributions that define them. We also discuss efficient choices of recycling distributions, both in terms of estimator efficiency and simulation efficiency.     

A recentred bootstrap procedure for constructing uniformly correct confidence sets under smooth function models

It has been found, under a smooth function model setting, that the n out of n bootstrap is inconsistent at stationary points of the smooth function, but that the m out of n bootstrap is consistent, provided that a correct convergence rate is specified of the plug-in smooth function estimator. By considering a more general moving-parameter framework, we show that neither of the above bootstrap methods is consistent uniformly over neighbourhoods of stationary points, so that anomalies often arise of coverages of bootstrap sets over certain subsets of parameter values. We propose a recentred bootstrap procedure for constructing confidence sets with uniformly correct coverages over compact sets containing stationary points. A weighted bootstrap procedure is also proposed as an alternative under more general circumstances. Unlike the m out of n bootstrap, both procedures do not require knowledge of the convergence rate of the smooth function estimator. Empirical performance of our procedures is illustrated with numerical examples.     

Two new data-dependent choices of m when applying the m-out-of-n bootstrap to hypothesis testing

The traditional non-parametric bootstrap (referred to as the n-out-of-n bootstrap) is a widely applicable and powerful tool for statistical inference, but in important situations it can fail. It is well known that by using a bootstrap sample of size m, different from n, the resulting m-out-of-n bootstrap provides a method for rectifying the traditional bootstrap inconsistency. Moreover, recent studies have shown that interesting cases exist where it is better to use the m-out-of-n bootstrap in spite of the fact that the n-out-of-n bootstrap works. In this paper, we discuss another case by considering its application to hypothesis testing. Two new data-based choices of m are proposed in this set-up. The results of simulation studies are presented to provide empirical comparisons between the performance of the traditional bootstrap and the m-out-of-n bootstrap, based on the two data-dependent choices of m, as well as on an existing method in the literature for choosing m. These results show that the m-out-of-n bootstrap, based on our choice of m, generally outperforms the traditional bootstrap procedure as well as the procedure based on the choice of m proposed in the literature.     

FAST DOUBLE BOOTSTRAP TESTS OF NONNESTED LINEAR REGRESSION MODELS

ABSTRACT

It has been shown in previous work that bootstrapping the J test for nonnested linear regression models dramatically improves its finite-sample performance. We provide evidence that a more sophisticated bootstrap procedure, which we call the fast double bootstrap, produces a very substantial further improvement in cases where the ordinary bootstrap does not work as well as it might. This FDB procedure is only about twice as expensive as the usual single bootstrap.     

Frequency domain bootstrap for ratio statistics under long-range dependence

A frequency domain bootstrap (FDB) is a common technique to apply Efron’s independent and identically distributed resampling technique (Efron, 1979) to periodogram ordinates – especially normalized periodogram ordinates – by using spectral density estimates. The FDB method is applicable to several classes of statistics, such as estimators of the normalized spectral mean, the autocorrelation (but not autocovariance), the normalized spectral density function, and Whittle parameters. While this FDB method has been extensively studied with respect to short-range dependent time processes, there is a dearth of research on its use with long-range dependent time processes. Therefore, we propose an FDB methodology for ratio statistics under long-range dependence, using semi- and nonparametric spectral density estimates as a normalizing factor. It is shown that the FDB approximation allows for valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without any stringent assumptions on the distribution of the underlying process. The results of a large simulation study show that the FDB approximation using a semi- or nonparametric spectral density estimator is often robust for various values of a long-memory parameter reflecting magnitude of dependence. We apply the proposed procedure to two data examples.     

Comparison of bootstrap and generalized bootstrap methods for estimating high quantiles

The generalized bootstrap is a parametric bootstrap method in which the underlying distribution function is estimated by fitting a generalized lambda distribution to the observed data. In this study, the generalized bootstrap is compared with the traditional parametric and non-parametric bootstrap methods in estimating the quantiles at different levels, especially for high quantiles. The performances of the three methods are evaluated in terms of cover rate, average interval width and standard deviation of width of the 95% bootstrap confidence intervals. Simulation results showed that the generalized bootstrap has overall better performance than the non-parametric bootstrap in high quantile estimation.     

Test for the existence of finite moments via bootstrap

This paper develops a bootstrap hypothesis test for the existence of finite moments of a random variable, which is nonparametric and applicable to both independent and dependent data. The test is based on a property in bootstrap asymptotic theory, in which the m out of n bootstrap sample mean is asymptotically normal when the variance of the observations is finite. Consistency of the test is established. Monte Carlo simulations are conducted to illustrate the finite sample performance and compare it with alternative methods available in the literature. Applications to financial data are performed for illustration.     

On a class of m out of n bootstrap confidence intervals

It is widely known that bootstrap failure can often be remedied by using a technique known as the ' m out of n ' bootstrap, by which a smaller number, m say, of observations are resampled from the original sample of size n . In successful cases of the bootstrap, the m out of n bootstrap is often deemed unnecessary. We show that the problem of constructing nonparametric confidence intervals is an exceptional case. By considering a new class of m out of n bootstrap confidence limits, we develop a computationally efficient approach based on the double bootstrap to construct the optimal m out of n bootstrap intervals. We show that the optimal intervals have a coverage accuracy which is comparable with that of the classical double-bootstrap intervals, and we conduct a simulation study to examine their performance. The results are in general very encouraging. Alternative approaches which yield even higher order accuracy are also discussed.     

A block bootstrap comparison for sparse chains

In this paper we apply the sequential bootstrap method proposed by Collet et al. [Bootstrap Central Limit theorem for chains of infinite order via Markov approximations, Markov Processes and Related Fields 11(3) (2005), pp. 443–464] to estimate the variance of the empirical mean of a special class of chains of infinite order called sparse chains. For this process, we show that we are able to compute numerically the true value of the standard error with any fixed error.

Our main goal is to present a comparison, for sparse chains, among sequential bootstrap, the block bootstrap method proposed by Künsch [The jackknife and the Bootstrap for general stationary observations, Ann. Statist. 17 (1989), pp. 1217–1241] and improved by Liu and Singh [Moving blocks jackknife and Bootstrap capture week dependence, in Exploring the limits of the Bootstrap, R. Lepage and L. Billard, eds., Wiley, New York, 1992, pp. 225–248] and the bootstrap method proposed by Bühlmann [Blockwise bootstrapped empirical process for stationary sequences, Ann. Statist. 22 (1994), pp. 995–1012].     

Assessment of individual bioequivalence using sufficient bootstrap procedure

This paper proposes a sufficient bootstrap method, which uses only the unique observations in the resamples, to assess the individual bioequivalence under 2 × 4 randomized crossover design. The finite sample performance of the proposed method is illustrated by extensive Monte Carlo simulations as well as a real‐experimental data set, and the results are compared with those obtained by the traditional bootstrap technique. Our records reveal that the proposed method is a good competitor or even better than the classical percentile bootstrap confidence limits.     

New block bootstrap methods: Sufficient and/or ordered

In this study, we propose sufficient time series bootstrap methods that achieve better results than conventional non-overlapping block bootstrap, but with less computing time and lower standard errors of estimation. Also, we propose using a new technique using ordered bootstrapped blocks, to better preserve the dependency structure of the original data. The performance of the proposed methods are compared in a simulation study for MA(2) and AR(2) processes and in an example. The results show that our methods are good competitors that often exhibit improved performance over the conventional block methods.     

Obtaining prediction intervals for FARIMA processes using the sieve bootstrap

The sieve bootstrap (SB) prediction intervals for invertible autoregressive moving average (ARMA) processes are constructed using resamples of residuals obtained by fitting a finite degree autoregressive approximation to the time series. The advantage of this approach is that it does not require the knowledge of the orders, p and q, associated with the ARMA(p, q) model. Up until recently, the application of this method has been limited to ARMA processes whose autoregressive polynomials do not have fractional unit roots. The authors, in a 2012 publication, introduced a version of the SB suitable for fractionally integrated autoregressive moving average (FARIMA (p,d,q)) processes with 0<d<0.5 and established its asymptotic validity. Herein, we study the finite sample properties this new method and compare its performance against an older method introduced by Bisaglia and Grigoletto in 2001. The sieve bootstrap (SB) method is a numerically simpler alternative to the older method which requires the estimation of p, d, and q at every bootstrap step. Monte-Carlo simulation studies, carried out under the assumption of normal, mixture of normals, and exponential distributions for the innovations, show near nominal coverages for short-term and long-term SB prediction intervals under most situations. In addition, the sieve bootstrap method yields better coverage and narrower intervals compared to the Bisaglia–Grigoletto method in some situations, especially when the error distribution is a mixture of normals.     

Control variates and importance sampling for efficient bootstrap simulations

Importance sampling and control variates have been used as variance reduction techniques for estimating bootstrap tail quantiles and moments, respectively. We adapt each method to apply to both quantiles and moments, and combine the methods to obtain variance reductions by factors from 4 to 30 in simulation examples.We use two innovations in control variates—interpreting control variates as a re-weighting method, and the implementation of control variates using the saddlepoint; the combination requires only the linear saddlepoint but applies to general statistics, and produces estimates with accuracy of order n ^-1/2 B ^-1, where n is the sample size and B is the bootstrap sample size.We discuss two modifications to classical importance sampling—a weighted average estimate and a mixture design distribution. These modifications make importance sampling robust and allow moments to be estimated from the same bootstrap simulation used to estimate quantiles.     

A weighted bootstrap approach to bootstrap iteration

The operation of resampling from a bootstrap resample, encountered in applications of the double bootstrap, maybe viewed as resampling directly from the sample but using probability weights that are proportional to the numbers of times that sample values appear in the resample. This suggests an approximate approach to double-bootstrap Monte Carlo simulation, where weighted bootstrap methods are used to circumvent much of the labour involved in compounded Monte Carlo approximation. In the case of distribution estimation or, equivalently, confidence interval calibration, the new method may be used to reduce the computational labour. Moreover, the method produces the same order of magnitude of coverage error for confidence intervals, or level error for hypothesis tests, as a full application of the double bootstrap.