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.     

Stationary bootstrapping for realized covariations of high frequency financial data

This paper studies the stationary bootstrap applicability for realized covariations of high frequency asynchronous financial data. The stationary bootstrap method, which is characterized by a block-bootstrap with random block length, is applied to estimate the integrated covariations. The bootstrap realized covariance, bootstrap realized regression coefficient and bootstrap realized correlation coefficient are proposed, and the validity of the stationary bootstrapping for them is established both for large sample and for finite sample. Consistencies of bootstrap distributions are established, which provide us valid stationary bootstrap confidence intervals. The bootstrap confidence intervals do not require a consistent estimator of a nuisance parameter arising from nonsynchronous unequally spaced sampling while those based on a normal asymptotic theory require a consistent estimator. A Monte-Carlo comparison reveals that the proposed stationary bootstrap confidence intervals have better coverage probabilities than those based on normal approximation.     

Bootstrapping time series models

This paper surveys recent development in bootstrap methods and the modifications needed for their applicability in time series models. The paper discusses some guidelines for empirical researchers in econometric analysis of time series. Different sampling schemes for bootstrap data generation and different forms of bootstrap test statistics are discussed. The paper also discusses the applicability of direct bootstrapping of data in dynamic models and cointegrating regression models. It is argued that bootstrapping residuals is the preferable approach. The bootstrap procedures covered include the recursive bootstrap, the moving block bootstrap and the stationary bootstrap.     

Bootstrapping time series models

This paper surveys recent development in bootstrap methods and the modifications needed for their applicability in time series models. The paper discusses some guidelines for empirical researchers in econometric analysis of time series. Different sampling schemes for bootstrap data generation and different forms of bootstrap test statistics are discussed. The paper also discusses the applicability of direct bootstrapping of data in dynamic models and cointegrating regression models. It is argued that bootstrapping residuals is the preferable approach. The bootstrap procedures covered include the recursive bootstrap, the moving block bootstrap and the stationary bootstrap.     

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.     

Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions

One of the indicators for evaluating the capability of a process is the process capability index. In this article, bootstrap confidence intervals of the generalized process capability index (GPCI) proposed by Maiti et al. are studied through simulation, when the underlying distributions are Lindley and Power Lindley distributions. The maximum likelihood method is used to estimate the parameters of the models. Three bootstrap confidence intervals namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile bootstrap (BCPB) are considered for obtaining confidence intervals of GPCI. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average width of the bootstrap confidence intervals. Simulation results show that the estimated coverage probabilities of the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval get closer to the nominal confidence level than those of the standard bootstrap confidence interval. Finally, three real datasets are analyzed for illustrative purposes.     

Comparing three bootstrap methods for survey data

Various bootstrap methods for variance estimation and confidence intervals in complex survey data, where sampling is done without replacement, have been proposed in the literature. The oldest, and perhaps the most intuitively appealing, is the without-replacement bootstrap (BWO) method proposed by Gross (1980). Unfortunately, the BWO method is only applicable to very simple sampling situations. We first introduce extensions of the BWO method to more complex sampling designs. The performance of the BWO and two other bootstrap methods, the rescaling bootstrap (Rao and Wu 1988) and the mirror-match bootstrap (Sitter 1992), are then compared through a simulation study. Together these three methods encompass the various bootstrap proposals.     

Bootstrap and Bayesian bootstrap clones for censored Markov chains

This is a study of the behaviors of the naive bootstrap and the Bayesian bootstrap clones designed to approximate the sampling distribution of the Aalen–Johansen estimator of a non-homogeneous censored Markov chain. The study shows that the approximations based on the Bayesian bootstrap clones and the naive bootstrap are first-order asymptotically equivalent. The two bootstrap methods are illustrated by a marketing example, and their performance is validated by a Monte Carlo experiment.     

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.     

The exact bootstrap mean and variance of an L-estimator

Exact analytic expressions for the bootstrap mean and variance of any L -estimator are obtained, thus eliminating the error due to bootstrap resampling. The expressions follow from the direct calculation of the bootstrap mean vector and covariance matrix of the whole set of order statistics. By using these expressions, recommendations can be made about the appropriateness of bootstrap estimation under given conditions.     

协整参数的自举推断

 针对完全修正最小二乘(full-modified ordinary least square,简称FMOLS）估计方法,给出一种协整参数的自举推断程序,证明零假设下自举统计量与检验统计量具有相同的渐近分布。关于检验功效的研究表明,虽然有约束自举的实际检验水平表现良好,但如果零假设不成立,自举统计量的分布是不确定的,因而其经验分布不能作为检验统计量精确分布的有效估计。实际应用中建议使用无约束自举,因为无论观测数据是否满足零假设,其自举统计量与零假设下检验统计量都具有相同的渐近分布。最后,利用蒙特卡洛模拟对自举推断和渐近推断的有限样本表现进行比较研究。     

Nonparametric confidence regions via the analytic wild bootstrap

The wild bootstrap is a nonparametric tool that can be used to estimate a sampling distribution in the presence of heteroscedastic errors. In particular, the wild bootstrap enables us to compute confidence regions for regression parameters under non-i.i.d. models. While the wild bootstrap may perform well in these settings, its obvious drawback is a lack of computational efficiency. The wild bootstrap requires a large number of bootstrap replications, making the use of this tool impractical when dealing with big data. We introduce the analytic wild bootstrap (ANWB), which provides a nonparametric alternative way of constructing confidence regions for regression parameters. The ANWB is superior to the wild bootstrap from a computational standpoint while exhibiting similar finite-sample performance. We report simulation results for both least squares and ridge regression. Additionally, we test the ANWB on a real dataset and compare its performance with that of other standard approaches.     

Bayesian bootstrap prediction

In this paper, bootstrap prediction is adapted to resolve some problems in small sample datasets. The bootstrap predictive distribution is obtained by applying Breiman's bagging to the plug-in distribution with the maximum likelihood estimator. The effectiveness of bootstrap prediction has previously been shown, but some problems may arise when bootstrap prediction is constructed in small sample datasets. In this paper, Bayesian bootstrap is used to resolve the problems. The effectiveness of Bayesian bootstrap prediction is confirmed by some examples. These days, analysis of small sample data is quite important in various fields. In this paper, some datasets are analyzed in such a situation. For real datasets, it is shown that plug-in prediction and bootstrap prediction provide very poor prediction when the sample size is close to the dimension of parameter while Bayesian bootstrap prediction provides stable prediction.     

EMPIRICAL LIKELIHOOD METHODS FOR THE GINI INDEX

The Gini index and its generalizations have been used extensively for measuring inequality and poverty in the social sciences. Recently, interval estimation based on nonparametric statistics has been proposed in the literature, for example the naive bootstrap method, the iterated bootstrap method and the bootstrap method via a pivotal statistic. In this paper, we propose empirical likelihood methods to construct confidence intervals for the Gini index or the difference of two Gini indices. Simulation studies show that the proposed empirical likelihood method performs slightly worse than the bootstrap method based on a pivotal statistic in terms of coverage accuracy, but it requires less computation. However, the bootstrap calibration of the empirical likelihood method performs better than the bootstrap method based on a pivotal statistic.     

Simultaneous bootstrap for all three parameters in random coefficient autoregressive models

In this paper we consider autoregressive processes with random coefficients and develop bootstrap approaches that asymptotically work for the distribution of estimated autoregressive parameter as well as for the distribution of estimated variances of the innovation noise and the disturbance noise. We discuss how to obtain approximative residuals of the process and how to separate between the innovation and the disturbance noise in order to be able to extend the classical residual bootstrap for autoregressive processes to the situation considered in this paper. Thereafter, we propose a wild bootstrap procedure as a variation of the residual bootstrap that uses estimated densities of the innovation and the disturbance noise to generate bootstrap replicates of the data generating process. The consistency of the bootstrap approaches is established and their performance is illustrated by a simulation study.     

Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

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.     

Bootstrapping with auxiliary information

In the nonparametric setting, the standard bootstrap method is based on the empirical distribution function of a random sample. The author proposes, by means of the empirical likelihood technique, an alternative bootstrap procedure under a nonparametric model in which one has some auxiliary information about the population distribution. By proving the almost sure weak convergence of the modified bootstrapped empirical process, the validity of the proposed bootstrap procedure is established. This new result is used to obtain bootstrap confidence bands for the population distribution function and to perform the bootstrap Kolmogorov test in the presence of auxiliary information. Other applications include bootstrapping means and variances with auxiliary information. Three simulation studies are presented to demonstrate the performance of the proposed bootstrap procedure for small samples.     

A comparison of bootstrap approaches for estimating uncertainty of parameters in linear mixed‐effects models

A version of the nonparametric bootstrap, which resamples the entire subjects from original data, called the case bootstrap, has been increasingly used for estimating uncertainty of parameters in mixed‐effects models. It is usually applied to obtain more robust estimates of the parameters and more realistic confidence intervals (CIs). Alternative bootstrap methods, such as residual bootstrap and parametric bootstrap that resample both random effects and residuals, have been proposed to better take into account the hierarchical structure of multi‐level and longitudinal data. However, few studies have been performed to compare these different approaches. In this study, we used simulation to evaluate bootstrap methods proposed for linear mixed‐effect models. We also compared the results obtained by maximum likelihood (ML) and restricted maximum likelihood (REML). Our simulation studies evidenced the good performance of the case bootstrap as well as the bootstraps of both random effects and residuals. On the other hand, the bootstrap methods that resample only the residuals and the bootstraps combining case and residuals performed poorly. REML and ML provided similar bootstrap estimates of uncertainty, but there was slightly more bias and poorer coverage rate for variance parameters with ML in the sparse design. We applied the proposed methods to a real dataset from a study investigating the natural evolution of Parkinson's disease and were able to confirm that the methods provide plausible estimates of uncertainty. Given that most real‐life datasets tend to exhibit heterogeneity in sampling schedules, the residual bootstraps would be expected to perform better than the case bootstrap. Copyright © 2013 John Wiley & Sons, Ltd.     

Extending the family of Bayesian bootstraps and exchangeable urn schemes

A Bayesian bootstrap for a finite population with censored observations is introduced. It is shown to reduce to the finite population Bayesian bootstrap if there is no censoring and to reduce to the censored data Bayesian bootstrap for a large population. A class of general urn schemes for simulating exchangeable sequences of variables is introduced which is connected to the bootstrap method.