Regeneration-based bootstrap for Markov chains

We consider a bootstrap method for Markov chains where the original chain is broken into a (random) number of cycles based on an atom (regeneration point) and the bootstrap scheme resamples from these cycles. We investigate the asymptotic accuracy of this method for the case of a sum (or a sample mean) related to the Markov chain. Under some standard moment conditions, the method is shown to be at least as good as the normal approximation, and better (second-order accurate) in the case of nonlattice summands. We give three examples to illustrate the applicability of our results.     

Asymptotic bootstrap validity for finite markov chains

Two methods of bootstrap, viz., standard, and conditional, are presented for estimating the transition probabilities of a finite state Markov chain. Asymptotic validity of the bootstrap estimates are established for both methods. An applica- tion to a bootstrapped statistic for testing independence is briefly discussed together with some simulation results.     

Robust bootstrap forecast densities for GARCH returns and volatilities

Bootstrap procedures are useful to obtain forecast densities for both returns and volatilities in the context of generalized autoregressive conditional heteroscedasticity models. In this paper, we analyse the effect of additive outliers on the finite sample properties of these bootstrap densities and show that, when obtained using maximum likelihood estimates of the parameters and standard filters for the volatilities, they are badly affected with dramatic consequences on the estimation of Value-at-Risk. We propose constructing bootstrap densities for returns and volatilities using a robust parameter estimator based on variance targeting implemented together with an adequate modification of the volatility filter. We show that the performance of the proposed procedure is adequate when compared with available robust alternatives. The results are illustrated with both simulated and real data.     

LASSO order selection for sparse autoregression: a bootstrap approach

Autoregressive models are widely employed for predictions and other inferences in many scientific fields. While the determination of their order is in general a difficult and critical step, this task becomes more complicated and crucial when the time series under investigation is realization of a stochastic process characterized by sparsity. In this paper we present a method for order determination of a stationary AR model with a sparse structure, given a set of observations, based upon a bootstrapped version of MAICE procedure [Akaike H. Prediction and entropy. Springer; 1998], in conjunction with a LASSO-type constraining procedure for lag suppression of insignificant lags. Empirical results will be obtained via Monte Carlo simulations. The quality of our method is assessed by comparison with the commonly adopted cross-validation approach and the non bootstrap counterpart of the presented procedure.     

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.     

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.     

Prediction intervals for time series models with trend via sieve bootstrap

This paper discusses method for constructing the prediction intervals for time series model with trend using the sieve bootstrap procedure. Gasser–Müller type of kernel estimator is used for trend estimation and prediction. The boundary modification of the kernel is applied to control the edge effect and to construct the predictor of a trend.     

Diagnostics for the bootstrap and fast double bootstrap

ABSTRACT

The bootstrap is typically less reliable in the context of time-series models with serial correlation of unknown form than when regularity conditions for the conventional IID bootstrap apply. It is, therefore, useful to have diagnostic techniques capable of evaluating bootstrap performance in specific cases. Those suggested in this paper are closely related to the fast double bootstrap (FDB) and are not computationally intensive. They can also be used to gauge the performance of the FDB itself. Examples of bootstrapping time series are presented, which illustrate the diagnostic procedures, and show how the results can cast light on bootstrap performance.     

A simple bootstrap bandwidth selector for local polynomial fitting

A new, fully data-driven bandwidth selector with a double smoothing (DS) bias term and a data-driven variance estimator is developed following the bootstrap idea. The data-driven variance estimation does not involve any additional bandwidth selection. The proposed bandwidth selector convergences faster than a plug-in one due to the DS bias estimate, whereas the data-driven variance improves its finite sample performance clearly and makes it stable. Asymptotic results of the proposals are obtained. A comparative simulation study was done to show the overall gains and the gains obtained by improving either the bias term or the variance estimate, respectively. It is shown that the use of a good variance estimator is more important when the sample size is relatively small.     

A modified bootstrap estimator for the mean of an asymmetric distribution

A modified bootstrap estimator of the population mean is proposed which is a convex combination of the sample mean and sample median, where the weights are random quantities. The estimator is shown to be strongly consistent and asymptotically normally distributed. The small- and moderate-sample-size behavior of the estimator is investigated and compared with that of the sample mean by means of Monte Carlo studies. It is found that the newly proposed estimator has much smaller mean squared errors and also yields significantly shorter confidence intervals for the population mean.     

A comparison of tests of homogeneity for sparse contingency tables

Five tests of homogeneity for a 2x(k+l) contingency table are compared using Monte Carlo techniques. For these studiesit is assumed that k becomes large in such a way that thecontingency table is sparse for 2xk of the cells, but the sample size in two of the cells remains large. The test statistics studied are: the chi-square approximation to the Pearson test statistic, the chi-square approximation to the likelihood ratio statistic, the normal approximation to Zelterman's (1984)the normal approximation to Pearson's chi-square, and the normal approximation to the likelihood ratio statistic. For the range of parameters studied the chi-square approximation to Pearson's statistic performs consistently well with regard to its size and power.     

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.     

Efficient computation of the performance of bootstrap and jackknife estimators of the variance of L-statistics

Calculation of the bootstrap and the jackknife estimators of the variance of a statistic often relies on approximation techniques because the exact values are difficult if not impossible to obtain analytically. For the special case where the statistic is a linear combination of order statistics we propose to calculate the exact values combinatorically, thus completely eliminating the second-stage simulation error.     

Spatial estimation and rescaled spatial bootstrap approach for finite population

In this study, an attempt has been made to improve the sampling strategy incorporating spatial dependency at estimation stage considering usual aerial sampling scheme, such as simple random sampling, when the underlying population is finite and spatial in nature. Using the distances between spatial units, an improved method of estimation, viz. spatial estimation procedure, has been proposed for the estimation of finite population mean. Further, rescaled spatial bootstrap (RSB) methods have been proposed for approximately unbiased estimation of variance of the proposed spatial estimator (SE). The properties of the proposed SE and its corresponding RSB methods were studied empirically through simulation.     

Robust block bootstrap panel predictability tests*

This article develops two block bootstrap-based panel predictability test procedures that are valid under very general conditions. Some of the allowable features include cross-sectional dependence, heterogeneous predictive slopes, persistent predictors, and complex error dynamics, including cross-unit endogeneity. While the first test procedure tests if there is any predictability at all, the second procedure determines the units for which predictability holds in case of a rejection by the first. A weak unit root framework is adopted to allow persistent predictors, and a novel theory is developed to establish asymptotic validity of the proposed bootstrap. Simulations are used to evaluate the performance of our tests in small samples, and their implementation is illustrated through an empirical application to stock returns.     

ESTIMATING A PARAMETER WHEN IT IS KNOWN THAT THE PARAMETER EXCEEDS A GIVEN VALUE

In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘θ > θ₁ ’ or ‘θ < θ₁ ’, where θ₁ is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ₁ is to replace an estimator, , of θ by the maximum of and θ₁. However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ₁ , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ₁ , strictly exceed θ₁ except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ₁, and more generally produces estimators that respect the constraint in a smooth, realistic fashion.     

On the relative performance of the block bootstrap for dependent data

In the independent setting, both Efron's bootstrap and “empiricai Edgeworth expansion” (E.E-expansion) give second-order accurate approximations to distributions of standardized and studentized statistics in the smooth function model. As a result, Efron's bootstrap was often regarded as roughly equivalent to the one-term E.E-expansion. However, a more detailed analysis shows that Efron's bootstrap outperforms the E.E-expansion in terms of loss functions by Bhattacharya and Qumsiyeh (1989) and in terms of probabilities for large deviations by Hall (1990) and Jing et a1 (1994). in this paper, we shall study the performances of the block bootstrap and the E.E-expansion for the weakly dependent data. It turns out that similar properties hold:both perform equally well at the center of the distribution but the block bootstrap provides accurate approximations even in the tails of the distributions. The study is focued on the simple case of standardized and studentized sample mean, but the conclusions can be easily extended to the smooth function of multivariate means.     

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.     

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.