Bootstrap confidence intervals in mixtures of discrete distributions

The problem of building bootstrap confidence intervals for small probabilities with count data is addressed. The law of the independent observations is assumed to be a mixture of a given family of power series distributions. The mixing distribution is estimated by nonparametric maximum likelihood and the corresponding mixture is used for resampling. We build percentile-t and Efron percentile bootstrap confidence intervals for the probabilities and we prove their consistency in probability. The new theoretical results are supported by simulation experiments for Poisson and geometric mixtures. We compare percentile-t and Efron percentile bootstrap intervals with eight other bootstrap or asymptotic theory based intervals. It appears that Efron percentile bootstrap intervals outperform the competitors in terms of coverage probability and length.     

Simultaneous bootstrap confidence bands in regression

We suggest pivotal methods for constructing simultaneous bootstrap confidence bands in regression. Most attention is given to the problem of simple linear regression, but our techniques admit trivial extension to other cases, including polynomial regression. The advantages of our bootstrap approach are twofold. Firstly, the bootstrap allows a very general distribution for the errors, and secondly, it admits a wide variety of shapes for the confidence band. In our technique the shape of each envelope of the band is determined by a general template, chosen by the experimenter, and bootstrap methods are used to select the scale of the template.     

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.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.     

A bias correction and acceleration approach for the problem of regions

For testing the problem of regions in the space of distribution functions, this paper considers approaches to modify the bootstrap probability to be a second-order accurate p$p$-value based on the familiar bias correction and acceleration method. It is shown that Shimodaira's [2004a. Approximately unbiased tests of regions using multistep-multiscale bootstrap resampling. Ann. Statist. 32, 2616–2641] twostep-multiscale bootstrap method works even in the problem of regions in functional space. In this paper the bias correction quantity is estimated by his onestep-multiscale bootstrap method. Instead of using the twostep-multiscale bootstrap method, the acceleration constant is estimated by a newly proposed jackknife method which requires first-level bootstrap resamplings only. Some numerical examples are illustrated, in which an application to testing significance in model selection is included.     

Total least squares and bootstrapping with applications in calibration

The solution to the errors-in-variables problem computed through total least squares is highly nonlinear. Because of this, many statistical procedures for constructing confidence intervals and testing hypotheses cannot be applied. One possible solution to this dilemma is bootstrapping. A nonparametric bootstrap technique could fail. Here, the proper nonparametric bootstrap procedure is provided and its correctness is proved. On the other hand, a residual bootstrap is not valid and suitable in this case. The results are illustrated through a simulation study. An application of this approach to calibration data is presented.     

Double bootstrap estimation of variance under systematic sampling with probability proportional to size

Variance estimation under systematic sampling with probability proportional to size is known to be a difficult problem. We attempt to tackle this problem by the bootstrap resampling method. It is shown that the usual way to bootstrap fails to give satisfactory variance estimates. As a remedy, we propose a double bootstrap method which is based on certain working models and involves two levels of resampling. Unlike existing methods which deal exclusively with the Horvitz–Thompson estimator, the double bootstrap method can be used to estimate the variance of any statistic. We illustrate this within the context of both mean and median estimation. Empirical results based on five natural populations are encouraging.     

The estimating function bootstrap

The authors propose a bootstrap procedure which estimates the distribution of an estimating function by resampling its terms using bootstrap techniques. Studentized versions of this so‐called estimating function (EF) bootstrap yield methods which are invariant under reparametrizations. This approach often has substantial advantage, both in computation and accuracy, over more traditional bootstrap methods and it applies to a wide class of practical problems where the data are independent but not necessarily identically distributed. The methods allow for simultaneous estimation of vector parameters and their components. The authors use simulations to compare the EF bootstrap with competing methods in several examples including the common means problem and nonlinear regression. They also prove symptotic results showing that the studentized EF bootstrap yields higher order approximations for the whole vector parameter in a wide class of problems.     

Bootstrap adaptive estimation: The trimmed-mean example

We consider the problem of choosing among a class of possible estimators by selecting the estimator with the smallest bootstrap estimate of finite sample variance. This is an alternative to using cross-validation to choose an estimator adaptively. The problem of a confidence interval based on such an adaptive estimator is considered. We illustrate the ideas by applying the method to the problem of choosing the trimming proportion of an adaptive trimmed mean. It is shown that a bootstrap adaptive trimmed mean is asymptotically normal with an asymptotic variance equal to the smallest among trimmed means. The asymptotic coverage probability of a bootstrap confidence interval based on such adaptive estimators is shown to have the nominal level. The intervals based on the asymptotic normality of the estimator share the same asymptotic result, but have poor small-sample properties compared to the bootstrap intervals. A small-sample simulation demonstrates that bootstrap adaptive trimmed means adapt themselves rather well even for samples of size 10.     

Detection and estimation of structural change in heavy-tailed sequence

In this paper, bootstrap detection and ratio estimation are proposed to analysis mean change in heavy-tailed distribution. First, the test statistic is constructed into a ratio form on the CUSUM process. Then, the asymptotic distribution of test statistic is obtained and the consistency of the test is proved. To solve the problem that the null distribution of the test statistic contains unknown tail index, we present a bootstrap approximation method to determine the critical values of the null distribution. We also discuss how to estimate change point based on ratio method. The consistency and rate of convergence for the change-point estimator are established. Finally, the excellent performance of our method is demonstrated through simulations using artificial and real data sets. Especially the simulation results of bootstrap test are better than those of another existing method.     

The robustness, reliabiligy and power of heteroskedasticity tests

Several tests for heteroskedasticity in linear regression models are examined. Asymptoticrobustness to heterokurticity, nonnormality and skewness is discussed. The finite sample eliability of asymptotically valid tests is investigated using Monte Carlo experiments. It is found that asymptotic critical values cannot, in general. be relied upon to give good agreement between nominal and actual finite sample significance levels. The use of the bootstrap overcomes this problem for general approaches that lead to asymptotically pivotal test statistics. Power comparisons are made for bootstrap tests and modified Glejser and Koenker tests are recommended.     

Estimation of Standard Errors of Empirical Bayes Estimators in Capm-Type Models

This paper is concerned with the problem of estimating the standard errors of the empirical Bayes estimators in linear regression models. The problem of deriving an exact expression for the standard error of this estimator is generally intractable. We suggest a procedure based on Efron’s bootstrap method as a way of estimating the standard error. It is shown, through simulations, that the bootstrap method provides a more accurate estimate of the standard error of the empirical Bayes estimator than the traditional large sample method.     

IMPORTANCE BOOTSTRAP RESAMPLING FOR PROPORTIONAL HAZARDS REGRESSION

Importance resampling is an approach that uses exponential tilting to reduce the resampling necessary for the construction of nonparametric bootstrap confidence intervals. The properties of bootstrap importance confidence intervals are well established when the data is a smooth function of means and when there is no censoring. However, in the framework of survival or time-to-event data, the asymptotic properties of importance resampling have not been rigorously studied, mainly because of the unduly complicated theory incurred when data is censored. This paper uses extensive simulation to show that, for parameter estimates arising from fitting Cox proportional hazards models, importance bootstrap confidence intervals can be constructed if the importance resampling probabilities of the records for the n individuals in the study are determined by the empirical influence function for the parameter of interest. Our results show that, compared to uniform resampling, importance resampling improves the relative mean-squared-error (MSE) efficiency by a factor of nine (for n = 200). The efficiency increases significantly with sample size, is mildly associated with the amount of censoring, but decreases slightly as the number of bootstrap resamples increases. The extra CPU time requirement for calculating importance resamples is negligible when compared to the large improvement in MSE efficiency. The method is illustrated through an application to data on chronic lymphocytic leukemia, which highlights that the bootstrap confidence interval is the preferred alternative to large sample inferences when the distribution of a specific covariate deviates from normality. Our results imply that, because of its computational efficiency, importance resampling is recommended whenever bootstrap methodology is implemented in a survival framework. Its use is particularly important when complex covariates are involved or the survival problem to be solved is part of a larger problem; for instance, when determining confidence bounds for models linking survival time with clusters identified in gene expression microarray data.     

Applications of the Bootstrap in ROC Analysis

The problem of estimating standard errors for diagnostic accuracy measures might be challenging for many complicated models. We can address such a problem by using the Bootstrap methods to blunt its technical edge with resampled empirical distributions. We consider two cases where bootstrap methods can successfully improve our knowledge of the sampling variability of the diagnostic accuracy estimators. The first application is to make inference for the area under the ROC curve resulted from a functional logistic regression model which is a sophisticated modelling device to describe the relationship between a dichotomous response and multiple covariates. We consider using this regression method to model the predictive effects of multiple independent variables on the occurrence of a disease. The accuracy measures, such as the area under the ROC curve (AUC) are developed from the functional regression. Asymptotical results for the empirical estimators are provided to facilitate inferences. The second application is to test the difference of two weighted areas under the ROC curve (WAUC) from a paired two sample study. The correlation between the two WAUC complicates the asymptotic distribution of the test statistic. We then employ the bootstrap methods to gain satisfactory inference results. Simulations and examples are supplied in this article to confirm the merits of the bootstrap methods.     

Moment Consistency of the Exchangeably Weighted Bootstrap for Semiparametric M‐estimation

The bootstrap variance estimate is widely used in semiparametric inferences. However, its theoretical validity is a well‐known open problem. In this paper, we provide a first theoretical study on the bootstrap moment estimates in semiparametric models. Specifically, we establish the bootstrap moment consistency of the Euclidean parameter, which immediately implies the consistency of t‐type bootstrap confidence set. It is worth pointing out that the only additional cost to achieve the bootstrap moment consistency in contrast with the distribution consistency is to simply strengthen the L₁ maximal inequality condition required in the latter to the L_p maximal inequality condition for p≥1. The general L_p multiplier inequality developed in this paper is also of independent interest. These general conclusions hold for the bootstrap methods with exchangeable bootstrap weights, for example, non‐parametric bootstrap and Bayesian bootstrap. Our general theory is illustrated in the celebrated Cox regression model.     

Bootstrap-assisted tests of symmetry for dependent data*

The paper considers the problem of testing for symmetry (about an unknown centre) of the marginal distribution of a strictly stationary and weakly dependent stochastic process. The possibility of using the autoregressive sieve bootstrap and stationary bootstrap procedures to obtain critical values and P-values for symmetry tests is explored. Bootstrap-assisted tests for symmetry are straightforward to implement and require no prior estimation of asymptotic variances. The small-sample properties of a wide variety of tests are investigated using Monte Carlo experiments. A bootstrap-assisted version of the triples test is found to have the best overall performance.     

Efficient bootstrap methods: A review

Summary One of the fundamental of mathematical statistics is the estimation of sampling characteristics of a random variable, a problem that is increasingly solved using bootstrap methods. Often these involve Monte Carlo simulation, but they may be costly and time-consuming in certain problems. Various methods for reducing the simulation cost in bootstrap simulations have been proposed, most of them applicable to simple random samples. Here we review the literature on efficient resampling methods, make comparisons, try to assess the best method for a particular problem.     

Importance resampling for the smoothed bootstrap

Alternative methods of estimating properties of unknown distributions include the bootstrap and the smoothed bootstrap. In the standard bootstrap setting, Johns (1988) introduced an importance resam¬pling procedure that results in more accurate approximation to the bootstrap estimate of a distribution function or a quantile. With a suitable “exponential tilting” similar to that used by Johns, we derived a smoothed version of importance resampling in the framework of the smoothed bootstrap. Smoothed importance resampling procedures were developed for the estimation of distribution functions of the Studentized mean, the Studentized variance, and the correlation coefficient. Implementation of these procedures are presented via simulation results which concentrate on the problem of estimation of distribution functions of the Studentized mean and Studentized variance for different sample sizes and various pre-specified smoothing bandwidths for the normal data; additional simulations were conducted for the estimation of quantiles of the distribution of the Studentized mean under an optimal smoothing bandwidth when the original data were simulated from three different parent populations: lognormal, t(3) and t(10). These results suggest that in cases where it is advantageous to use the smoothed bootstrap rather than the standard bootstrap, the amount of resampling necessary might be substantially reduced by the use of importance resampling methods and the efficiency gains depend on the bandwidth used in the kernel density estimation.     

On the Bootstrap and Monotone Likelihood in the Cox Proportional Hazards Regression Model

Recent literature has provided encouragement for using the bootstrap for inference on regression parameters in the Cox proportional hazards (PH) model. However, generating and performing the necessary partial likelihood computations on multitudinous bootstrap samples greatly increases the chances of incurring problems with monotone likelihood at some point in the analysis. The only symptom of monotone likelihood may be a failure to converge in the numerical maximization procedure, and so the problem might naively be dismissed by deleting the offending data set and replacing it with a new one. This strategy is shown to lead to potentially high selection biases in the subsequent summary statistics. This note discusses the importance of keeping track of these monotone likelihood cases and provides recommendations for their use in interpreting bootstrap findings, and for avoiding unwanted biases that may result from high rates of occurrence. In many cases, high monotone likelihood rates indicate that a more highly-specified model may be preferred. Special consideration is given to the problem of high monotone likelihood incidence in Monte Carlo studies of the bootstrap.     

Bootstrap Testing for Changes in Persistence with Heavy-Tailed Innovations

This article considers the problem of detection for changes in persistence with heavy-tailed innovations. We adopt a ratio type test and derive its null asymptotic distribution which is dependent on the stable index. Then a residual-based bootstrap is proposed when the stable index is unknown. Our procedure requires drawing bootstrap samples of size m < T, T being the size of original sample. We establish the convergence in probability of the bootstrap distribution function assuming that m → ∞ and m/T → 0. A Monte Carlo study has shown that the bootstrap improve the finite sample size and power compared to the asymptotic test, especially for small stable index.