Improving the reliability of bootstrap tests with the fast double bootstrap

Two procedures are proposed for estimating the rejection probabilities (RPs) of bootstrap tests in Monte Carlo experiments without actually computing a bootstrap test for each replication. These procedures are only about twice as expensive (per replication) as estimating RPs for asymptotic tests. Then a new procedure is proposed for computing bootstrap P values that will often be more accurate than ordinary ones. This “fast double bootstrap” (FDB) is closely related to the double bootstrap, but it is far less computationally demanding. Simulation results for three different cases suggest that the FDB can be very useful in practice.     

Application of the bootstrap method for estimation of the quantile function

The bootstrap method was used to reduce the sample volume at estimating the quantile function. Accuracy of the quantile sample estimate vs. the distribution of random variable was established analytically. A numerical example of calculation of the quantiles using the proposed bootstrap procedure for the uniform, normal, and Cauchy distributions was considered. An approximate formula for calculation of the quantile of the normal distribution was determined.     

A time series bootstrap procedure for interpolation intervals

A sieve bootstrap procedure for constructing interpolation intervals for a general class of linear processes is proposed. This sieve bootstrap provides consistent estimators of the conditional distribution of the missing values, given the observed data. A Monte Carlo experiment is used to show the finite sample properties of the sieve bootstrap and finally, the performance of the proposed method is illustrated with a real data example.     

An empirical study of a test for polynomial relationships in randomly right censored regression models

In this paper, a test statistic is constructed to test polynomial relationships in randomly right censored regression models based on the local polynomial smoothing technique. Two bootstrap procedures, namely the residual-based bootstrap and the naive bootstrap procedures, are suggested to derive the p-value of the test. Some simulations are conducted to empirically assess the performance of the two bootstrap procedures. The results demonstrate that the residual-based bootstrap performs much better than the naive bootstrap and the test method with the residual-based bootstrap to derive the p-value works satisfactorily. Although the limiting distribution of the test statistic and the consistency of the bootstrap approximations remain to be investigated, simulation results indicate that the proposed test method may be of some practical use. As a real example, the proposed test is applied to the Stanford heart transplant data.     

Bootstrap and fast double bootstrap tests of cointegration rank with financial time series

The likelihood ratio test of cointegration rank is the most widely used test for cointegration. Many studies have shown that its finite sample distribution is not well approximated by the limiting distribution. Bootstrap and fast double bootstrap (FDB) algorithms for the likelihood ratio test are introduced and evaluated by Monte Carlo simulation experiments. It is found that the performance of the ordinary (single) bootstrap test is in most cases good in terms of the size of the test. The FDB produces a further improvement in cases where the performance of the asymptotic test is unsatisfactory and the single bootstrap test overrejects noticeably. The FDB is shown to be a useful supplement to the single bootstrap as a tool for determining the cointegration rank. The tests are applied to US interest rates and international stock prices series. By simulating the data assuming that the cointegration rank is known, it is found that the asymptotic test tends to overestimate the cointegration rank, while the bootstrap and FDB tests choose the correct cointegration rank.     

Modified fast double sieve bootstraps for ADF tests

This paper studies the finite sample performance of the sieve bootstrap augmented Dickey-Fuller (ADF) unit root test. It is well known that this test’s accuracy in terms of rejection probability under the null depends greatly on the underlying DGP. Through extensive simulations, we find that it also depends on the number of lags employed in the bootstrap DGP and in the bootstrap ADF regression. Based on this finding and using some well established theoretical results, we propose a simple modification that significantly improves the test’s accuracy. We also introduce different versions of the fast double bootstrap, each modified according to the same theoretical basis. According to our simulations, these new testing procedures have lower error in rejection probability under the null while retaining good power.     

Simple bootstrap statistical inference using the SAS system.

Nonparametric bootstrap statistical inference is a robust computer intensive method for generating estimates of statistical variability for which formulae are not known or asymptotic assumptions are not met. A SAS macro that implements simple nonparametric bootstrap statistical inference is presented with an example. The program code is easily generalized to any SAS procedure which includes a BY statement, and to cases of clustered data.     

On the use of the bootstrap for estimating functions with functional data

The bootstrap methodology for functional data and functional estimation target is considered. A Monte Carlo study analyzing the performance of the bootstrap confidence bands (obtained with different resampling methods) of several functional estimators is presented. Some of these estimators (e.g., the trimmed functional mean) rely on the use of depth notions for functional data and do not have received yet much attention in the literature. A real data example in cardiology research is also analyzed. In a more theoretical aspect, a brief discussion is given providing some insights on the asymptotic validity of the bootstrap methodology when functional data, as well as a functional parameter, are involved.     

Model based bootstrap methods for interval censored data

The performance of model based bootstrap methods for constructing point-wise confidence intervals around the survival function with interval censored data is investigated. It is shown that bootstrapping from the nonparametric maximum likelihood estimator of the survival function is inconsistent for the current status model. A model based smoothed bootstrap procedure is proposed and proved to be consistent. In fact, a general framework for proving the consistency of any model based bootstrap scheme in the current status model is established. In addition, simulation studies are conducted to illustrate the (in)-consistency of different bootstrap methods in mixed case interval censoring. The conclusions in the interval censoring model would extend more generally to estimators in regression models that exhibit non-standard rates of convergence.     

Weighted bootstrap method for fuzzy data

The weighted bootstrap contained in the monograph by Barbe and Bertail in Lecture Notes in Statist, Springer (1995) is a simple and straight-forward method for calculating approximated biases, standard deviations, confidence intervals, and so forth, in almost any nonparametric estimation problem. In this paper, we consider another example, namely, fuzzy data, and use the weighted bootstrap to answer several questions concerning the minimum inaccuracy estimator (Corral and Gil in Stochastica 8:63–81, 1984): (a) What is the standard error of this estimator? (b) What is a reasonable confidence interval for such a estimate? The validity of weighted bootstrap method is investigated using a real data and computer simulation.     

Creating non-parametric bootstrap samples using Poisson frequencies

This article describes how, in the high-level software packages used by non-statisticians, approximate non-parametric bootstrap samples can be created and analyzed without physically creating new data sets, or resorting to complex programming. The comparable performance of this shortcut method, which uses Poisson rather than multinomial frequencies for the numbers of copies of each observation, is demonstrated theoretically by evaluating the bootstrap variance in an example where the classic estimator of the sampling variance of the statistic of interest has a known closed form. For sample sizes of 50 or more, bootstrap standard errors obtained by this shortcut method exceeded those obtained by the standard version by less than 1%. The proposed method is also evaluated in two worked examples, involving statistics whose sampling distribution is more complex. The second of these is also used to illustrate when one can and cannot use non-parametric bootstrap samples.     

Bootstrap approach to the multi-sample test of means with imprecise data

A bootstrap approach to the multi-sample test of means for imprecisely valued sample data is introduced. For this purpose imprecise data are modelled in terms of fuzzy values. Populations are identified with fuzzy-valued random elements, often referred to in the literature as fuzzy random variables. An example illustrates the use of the suggested method. Finally, the adequacy of the bootstrap approach to test the multi-sample hypothesis of means is discussed through a simulation comparative study.     

On the efficiency of bootstrap method into the analysis contingency table

The bootstrap method is a computer intensive statistical method that is widely used in performing nonparametric inference. Categorical data analysis, in particular the analysis of contingency tables, is commonly used in applied field. This work considers nonparametric bootstrap tests for the analysis of contingency tables. There are only a few research papers which exploit this field. The p-values of tests in contingency tables are discrete and should be uniformly distributed under the null hypothesis. The results of this article show that corresponding bootstrap versions work better than the standard tests. Properties of the proposed tests are illustrated and discussed using Monte Carlo simulations. This article concludes with an analytical example that examines the performance of the proposed tests and the confidence interval of the association coefficient.     

地震信号去卷的一种自适应滤波方法

本文讨论地震信号去卷问题,提出了一种新的自适应递推去卷滤波器,它由参数和信号估 计的两段Bootstrap算法组成.其优点是:1)同增广状态Kalman滤波[2,3]相比,显著地减小 了计算量;2)采用了虚拟噪声补偿技术,有效地克服了滤波的发散.仿真例子说明了方法有效 性.     

The Numerical Performance of Fast Bootstrap Procedures

The numerical performance of faster ways to perform inference using thebootstrap are investigated. The bootstrap procedures are applied to tests foran unknown structural change and evaluated in a simulation environment. Thesimulation results indicate that these faster procedures work extremely well,and at a fraction of the computing costs. An empirical example illustrates theuse of fast bootstrap procedures that, this time, can be implemented by theapplied researcher.     

Multiple forecasts with autoregressive time series models: case studies

It is indisputable that accurate forecasts of economic activities are vital to successful business and government policies. In many circumstances, instead of a single forecast, simultaneous prediction intervals for multiple forecasts are more useful to decision-makers. For example, based on previous monthly sales records, a production manager would be interested in the next 12 interval forecasts of the monthly sales using for the annual inventory and manpower planning. For Gaussian autoregressive time series processes, several procedures for constructing simultaneous prediction intervals have been proposed in the literature. These methods assume a normal error distribution and can be adversely affected by departures from normality which are commonly encountered in business and economic time series. In this article, we explore the bootstrap methods for the construction of simultaneous multiple interval forecasts. To understand the mechanisms and characteristics of the proposed bootstrap procedures, several macro-economic time series are selected for illustrative purposes. The selected series are fitted reasonably well with autoregressive models which form an important class in time series. As a matter of fact, the major ideas discussed in this paper with autoregressive processes can be extended to other more complicated time series models.     

Improvement of time forecasting models using a novel hybridization of bootstrap and double bootstrap artificial neural networks

Hybrid models such as the Artificial Neural Network-Autoregressive Integrated Moving Average (ANN–ARIMA) model are widely used in forecasting. However, inaccuracies and inefficiency remain in evidence. To yield the ANN–ARIMA with a higher degree of accuracy, efficiency and precision, the bootstrap and the double bootstrap methods are commonly used as alternative methods through the reconstruction of an ANN–ARIMA standard error. Unfortunately, these methods have not been applied in time series-based forecasting models. The aims of this study are twofold. First, is to propose the hybridization of bootstrap model and that of double bootstrap mode called Bootstrap Artificial Neural Network-Autoregressive Integrated Moving Average (B-ANN–ARIMA) and Double Bootstrap Artificial Neural Network-Autoregressive Integrated Moving Average (DB-ANN–ARIMA), respectively. Second, is to investigate the performance of these proposed models by comparing them with ARIMA, ANN and ANN–ARIMA. Our investigation is based on three well-known real datasets, i.e., Wolf’s sunspot data, Canadian lynx data and, Malaysia ringgit/United States dollar exchange rate data. Statistical analysis on SSE, MSE, RMSE, MAE, MAPE and VAF is then conducted to verify that the proposed models are better than previous ARIMA, ANN and ANN–ARIMA models. The empirical results show that, compared with ARIMA, ANNs and ANN–ARIMA models, the proposed models generate smaller values of SSE, MSE, RMSE, MAE, MAPE and VAF for both training and testing datasets. In other words, the proposed models are better than those that we compare with. Their forecasting values are closer to the actual values. Thus, we conclude that the proposed models can be used to generate better forecasting values with higher degree of accuracy, efficiency and, precision in forecasting time series results becomes a priority.     

On properties of predictors derived with a two-step bootstrap model averaging approach—A simulation study in the linear regression model

In many applications of model selection there is a large number of explanatory variables and thus a large set of candidate models. Selecting one single model for further inference ignores model selection uncertainty. Often several models fit the data equally well. However, these models may differ in terms of the variables included and might lead to different predictions. To account for model selection uncertainty, model averaging procedures have been proposed. Recently, an extended two-step bootstrap model averaging approach has been proposed. The first step of this approach is a screening step. It aims to eliminate variables with negligible effect on the outcome. In the second step the remaining variables are considered in bootstrap model averaging. A large simulation study is performed to compare the MSE and coverage rate of models derived with bootstrap model averaging, the full model, backward elimination using Akaike and Bayes information criterion and the model with the highest selection probability in bootstrap samples. In a data example, these approaches are also compared with Bayesian model averaging. Finally, some recommendations for the development of predictive models are given.     

Local bootstrap approaches for fractional differential parameter estimation in ARFIMA models

In this paper we investigate bootstrap techniques applied to the estimation of the fractional differential parameter in ARFIMA models, d. The novelty is the focus on the local bootstrap of the periodogram function. The approach is then applied to three different semiparametric estimators of d, known from the literature, based upon the periodogram function. By means of an extensive set of simulation experiments, the bias and mean square errors are quantified for each estimator and the efficacy of the local bootstrap is stated in terms of low bias, short confidence intervals, and low CPU times. Finally, a real data set is analyzed to demonstrate that the methodology may be quite effective in solving real problems.     

Modified tests for variance changes in autoregressive regression

In this paper, we consider the problem of testing for variance changes in the linear autoregressive processes including AR(p) processes meanwhile autoregressive parameters shifts occur. In performing a test, we employ the conventional residual CUSUM of squares test (RCUSQ) statistic. The RCUSQ test is based on the bootstrap method introduced to eliminate the influence caused by the autoregressive parameters shifts. It is shown that under regularity conditions, the test statistic behaves asymptotically the function of a standard Brownian bridge. Simulation results as to AR(1) processes and an example of real data analysis are provided for illustration.