Bootstrapping moving average models

Summary In recent years, the bootstrap method has been extended to time series analysis where the observations are serially correlated. Contributions have focused on the autoregressive model producing alternative resampling procedures. In contrast, apart from some empirical applications, very little attention has been paid to the possibility of extending the use of the bootstrap method to pure moving average (MA) or mixed ARMA models. In this paper, we present a new bootstrap procedure which can be applied to assess the distributional properties of the moving average parameters estimates obtained by a least square approach. We discuss the methodology and the limits of its usage. Finally, the performance of the bootstrap approach is compared with that of the competing alternative given by the Monte Carlo simulation. Research partially supported by CNR and MURST.     

Small area estimation of poverty indicators

The authors propose to estimate nonlinear small area population parameters by using the empirical Bayes (best) method, based on a nested error model. They focus on poverty indicators as particular nonlinear parameters of interest, but the proposed methodology is applicable to general nonlinear parameters. They use a parametric bootstrap method to estimate the mean squared error of the empirical best estimators. They also study small sample properties of these estimators by model‐based and design‐based simulation studies. Results show large reductions in mean squared error relative to direct area‐specific estimators and other estimators obtained by “simulated” censuses. The authors also apply the proposed method to estimate poverty incidences and poverty gaps in Spanish provinces by gender with mean squared errors estimated by the mentioned parametric bootstrap method. For the Spanish data, results show a significant reduction in coefficient of variation of the proposed empirical best estimators over direct estimators for practically all domains. The Canadian Journal of Statistics 38: 369–385; 2010 © 2010 Statistical Society of Canada     

Robust unit root tests with autoregressive errors

ABSTRACT

This article presents a new test for unit roots based on least absolute deviation estimation specially designed to work for time series with autoregressive errors. The methodology used is a bootstrap scheme based on estimating a model and then the innovations. The resampling part is performed under the null hypothesis and, as it is customary in bootstrap procedures, is automatic and does not rely on the calculation of any nuisance parameter. The validity of the procedure is established and the asymptotic distribution of the statistic proposed is proved to converge to the correct distribution. To analyze the performance of the test for finite samples, a Monte Carlo study is conducted showing a very good behavior in many different situations.     

Bootstrapping a time series model: some empirical results

The bootstrap is a methodology for estimating standard errors. The idea is to use a Monte Carlo simulation experiment based on a nonparametric estimate of the error distribution. The main objective of this article is to demonstrate the use of the bootstrap to attach standard errors to coefficient estimates in a second-order autoregressive model fitted by least squares and maximum likelihood estimation. Additionally, a comparison of the bootstrap and the conventional methodology is made. As it turns out, the conventional asymptotic formulae (both the least squares and maximum likelihood estimates) for estimating standard errors appear to overestimate the true standard errors. But there are two problems:i. The first two observations y₁ and y₂ have been fixed, and ii. The residuals have not been inflated. After these two factors are considered in the trial and bootstrap experiment, both the conventional maximum likelihood and bootstrap estimates of the standard errors appear to be performing quite well.     

Estimating consumer acceptance limits

A methodology is developed for estimating consumer acceptance limits on a sensory attribute of a manufactured product. In concept these limits are analogous to engineering tolerances. The method is based on a generalization of Stevens' Power Law. This generalized law is expressed as a nonlinear statistical model. Instead of restricting the analysis to this particular case, a strategy is discussed for evaluating nonlinear models in general since scientific models are frequently of nonlinear form. The strategy focuses on understanding the geometrical contrasts between linear and nonlinear model estimation and assessing the bias in estimation and the departures from a Gaussian sampling distribution. Computer simulation is employed to examine the behavior of nonlinear least squares estimation. In addition to the usual Gaussian assumption, a bootstrap sample reuse procedure and a general triangular distribution are introduced for evaluating the effects of a non-Gaussian or asymmetrical error structure. Recommendations are given for further model analysis based on the simulation results. In the case of a model for which estimation bias is not a serious issue, estimating functions of the model are considered. Application of these functions to the generalization of Stevens’ Power Law leads to a means for defining and estimating consumer acceptance limits, The statistical form of the law and the model evaluation strategy are applied to consumer research data. Estimation of consumer acceptance limits is illustrated and discussed.     

PARTIALLY LINEAR MODEL SELECTION BY THE BOOTSTRAP

We propose a new approach to the selection of partially linear models based on the conditional expected prediction square loss function, which is estimated using the bootstrap. Because of the different speeds of convergence of the linear and the nonlinear parts, a key idea is to select each part separately. In the first step, we select the nonlinear components using an ' m -out-of- n ' residual bootstrap that ensures good properties for the nonparametric bootstrap estimator. The second step selects the linear components from the remaining explanatory variables, and the non-zero parameters are selected based on a two-level residual bootstrap. We show that the model selection procedure is consistent under some conditions, and our simulations suggest that it selects the true model most often than the other selection procedures considered.     

Evaluating the power of the mann-whitney test using the bootstrap method

In this paper we evaluate the power of the Mann-Whitney test in the shift model G(x) = F (x+θ) for all x , where the distribution of G is obtained by shifting F by an amount of θ.

The bootstrap method was used to evaluate the power of the Mann-Whitney test . A comparison among the bootstrap power , the asymptotic power of the Mann-Whitney test and the t-test power proved that the bootstrap is a better technique , because , it does not require the assumption of normality.     

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.     

Efficient mean estimation in log-normal linear models

Log-normal linear models are widely used in applications, and many times it is of interest to predict the response variable or to estimate the mean of the response variable at the original scale for a new set of covariate values. In this paper we consider the problem of efficient estimation of the conditional mean of the response variable at the original scale for log-normal linear models. Several existing estimators are reviewed first, including the maximum likelihood (ML) estimator, the restricted ML (REML) estimator, the uniformly minimum variance unbiased (UMVU) estimator, and a bias-corrected REML estimator. We then propose two estimators that minimize the asymptotic mean squared error and the asymptotic bias, respectively. A parametric bootstrap procedure is also described to obtain confidence intervals for the proposed estimators. Both the new estimators and the bootstrap procedure are very easy to implement. Comparisons of the estimators using simulation studies suggest that our estimators perform better than the existing ones, and the bootstrap procedure yields confidence intervals with good coverage properties. A real application of estimating the mean sediment discharge is used to illustrate the methodology.     

Model Selection,Transformations and Variance Estimation in Nonlinear Regression

The results of analyzing experimental data using a parametric model may heavily depend on the chosen model for regression and variance functions, moreover also on a possibly underlying preliminary transformation of the variables. In this paper we propose and discuss a complex procedure which consists in a simultaneous selection of parametric regression and variance models from a relatively rich model class and of Box-Cox variable transformations by minimization of a cross-validation criterion. For this it is essential to introduce modifications of the standard cross-validation criterion adapted to each of the following objectives: 1. estimation of the unknown regression function, 2. prediction of future values of the response variable, 3. calibration or 4. estimation of some parameter with a certain meaning in the corresponding field of application. Our idea of a criterion oriented combination of procedures (which usually if applied, then in an independent or sequential way) is expected to lead to more accurate results. We show how the accuracy of the parameter estimators can be assessed by a “moment oriented bootstrap procedure", which is an essential modification of the “wild bootstrap” of Härdle and Mammen by use of more accurate variance estimates. This new procedure and its refinement by a bootstrap based pivot (“double bootstrap”) is also used for the construction of confidence, prediction and calibration intervals. Programs written in Splus which realize our strategy for nonlinear regression modelling and parameter estimation are described as well. The performance of the selected model is discussed, and the behaviour of the procedures is illustrated, e.g., by an application in radioimmunological assay.     

The Third-Order Bias of Nonlinear Estimators

The third-order bias of nonlinear estimators is derived and illustrated using a variety of estimators popular in applied econometrics. A simulation using the exponential regression model indicates that the third-order analytical correction reduces bias substantially compared to higher-order bootstrap and Jackknife corrections, particularly in very small samples.     

A Bootstrap Algorithm for Data from a Periodic Multiplicative Intensity Function

The periodic multiplicative intensity model is considered. A new bootstrap method for non stationary counting processes which intensity function has some periodicity properties is presented. Its main advantage is that it does not destroy the temporal order and the original periodicity of the underlying counting process. The proposed algorithm is used to construct a bootstrap version of the maximum likelihood hazard function estimator. The consistency of the bootstrap method is shown. A possible modification of the proposed bootstrap method is discussed. The bootstrap simultaneous confidence intervals for the hazard function are presented. The telecommunication network traffic real data example is discussed.     

Resampling-based bias-corrected time series prediction

In this paper, we consider estimation of the mean squared prediction error (MSPE) of the best linear predictor of (possibly) nonlinear functions of finitely many future observations in a stationary time series. We develop a resampling methodology for estimating the MSPE when the unknown parameters in the best linear predictor are estimated. Further, we propose a bias corrected MSPE estimator based on the bootstrap and establish its second order accuracy. Finite sample properties of the method are investigated through a simulation study.     

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.     

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.     

Using bootstrap method to evaluate the estimates of nicotine equivalents from linear mixed model and generalized estimating equation

Twenty-four-hour urinary excretion of nicotine equivalents, a biomarker for exposure to cigarette smoke, has been widely used in biomedical studies in recent years. Its accurate estimate is important for examining human exposure to tobacco smoke. The objective of this article is to compare the bootstrap confidence intervals of nicotine equivalents with the standard confidence intervals derived from linear mixed model (LMM) and generalized estimation equation. We use percentile bootstrap method because it has practical value for real-life application and it works well with nicotine data. To preserve the within-subject correlation of nicotine equivalents between repeated measures, we bootstrap the repeated measures of each subject as a vector. The results indicate that the bootstrapped estimates in most cases give better estimates than the LMM and generalized estimation equation without bootstrap.     

An evaluation of bootstrap methods for outlier detection in least squares regression

Outlier detection is a critical part of data analysis, and the use of Studentized residuals from regression models fit using least squares is a very common approach to identifying discordant observations in linear regression problems. In this paper we propose a bootstrap approach to constructing critical points for use in outlier detection in the context of least-squares Studentized residuals, and find that this approach allows naturally for mild departures in model assumptions such as non-Normal error distributions. We illustrate our methodology through both a real data example and simulated data.     

A general methodology for bootstrapping in non-parametric frontier models

The Data Envelopment Analysis method has been extensively used in the literature to provide measures of firms' technical efficiency. These measures allow rankings of firms by their apparent performance. The underlying frontier model is non-parametric since no particular functional form is assumed for the frontier model. Since the observations result from some data-generating process, the statistical properties of the estimated efficiency measures are essential for their interpretations. In the general multi-output multi-input framework, the bootstrap seems to offer the only means of inferring these properties (i.e. to estimate the bias and variance, and to construct confidence intervals). This paper proposes a general methodology for bootstrapping in frontier models, extending the more restrictive method proposed in Simar & Wilson (1998) by allowing for heterogeneity in the structure of efficiency. A numerical illustration with real data is provided to illustrate the methodology.     

Bootstrap for U-statistics: a new approach

Bootstrap for nonlinear statistics like U-statistics of dependent data has been studied by several authors. This is typically done by producing a bootstrap version of the sample and plugging it into the statistic. We suggest an alternative approach of getting a bootstrap version of U-statistics. We will show the consistency of the new method and compare its finite sample properties in a simulation study and by applying both methods to financial data.     

Multiple Comparisons with More than One Control for Exponential Location Parameters Under Heteroscedasticity

In this article we present a simple bootstrap method for time series. The proposed method is model-free, and hence it enables us to avoid certain situations where the bootstrap samples may contain impossible values due to resampling from the residuals. The method is easy to implement and can be applied to stationary and nonstationary time series. The simulation results and the application to real time series data show that the method works very well.