Finite sample moments of a bootstrap estimator of the james-stein rule

The finite sample moments of the bootstrap estimator of the James-Stein rule are derived and shown to be biased. Analytical results shed some light upon the source of bias and suggest that the bootstrap will be biased in other settings where the moments of the statistic of interest depends on nonlinear functions of the parameters of its distribution.     

Yule–Walker type estimators in periodic bilinear models: strong consistency and asymptotic normality

This paper studies the covariance structure and the asymptotic properties of Yule–Walker (YW) type estimators for a bilinear time series model with periodically time-varying coefficients. We give necessary and sufficient conditions ensuring the existence of moments up to eighth order. Expressions of second and third order joint moments, as well as the limiting covariance matrix of the sample moments are given. Strong consistency and asymptotic normality of the YW estimator as well as hypotheses testing via Wald’s procedure are derived. We use a residual bootstrap version to construct bootstrap estimators of the YW estimates. Some simulation results will demonstrate the large sample behavior of the bootstrap procedure.     

A Framework for Random-Effects ROC Analysis: Biases with the Bootstrap and Other Variance Estimators

In this article, we analyze the three-way bootstrap estimate of the variance of the reader-averaged nonparametric area under the receiver operating characteristic (ROC) curve. The setting for this work is medical imaging, and the experimental design involves sampling from three distributions: a set of normal and diseased cases (patients), and a set of readers (doctors). The experiment we consider is fully crossed in that each reader reads each case. A reading generates a score that indicates the reader's level of suspicion that the patient is diseased. The distribution of scores for the normal patients is compared to the distribution of scores for the diseased patients via an ROC curve, and the area under the ROC curve (AUC) summarizes the reader's diagnostic ability to separate the normal patients from the diseased ones. We find that the bootstrap estimate of the variance of the reader-averaged AUC is biased, and we represent this bias in terms of moments of success outcomes. This representation helps unify and improve several current methods for multi-reader multi-case (MRMC) ROC analysis.     

Control variates and importance sampling for efficient bootstrap simulations

Importance sampling and control variates have been used as variance reduction techniques for estimating bootstrap tail quantiles and moments, respectively. We adapt each method to apply to both quantiles and moments, and combine the methods to obtain variance reductions by factors from 4 to 30 in simulation examples.We use two innovations in control variates—interpreting control variates as a re-weighting method, and the implementation of control variates using the saddlepoint; the combination requires only the linear saddlepoint but applies to general statistics, and produces estimates with accuracy of order n ^-1/2 B ^-1, where n is the sample size and B is the bootstrap sample size.We discuss two modifications to classical importance sampling—a weighted average estimate and a mixture design distribution. These modifications make importance sampling robust and allow moments to be estimated from the same bootstrap simulation used to estimate quantiles.     

Bootstrap-based bias corrections for INAR count time series

Integer-valued autoregressive (INAR) processes form a very useful class of processes suitable to model time series of counts. Several practically relevant estimators based on INAR data are known to be systematically biased away from their population values, e.g. sample autocovariances, sample autocorrelations, or the dispersion index. We propose to do bias correction for such estimators by using a recently proposed INAR-type bootstrap scheme that is tailor-made for INAR processes, and which has been proven to be asymptotically consistent under general conditions. This INAR bootstrap allows an implementation with and without parametrically specifying the innovations' distribution. To judge the potential of corresponding bias correction, we compare these bootstraps in simulations to several competitors that include the AR bootstrap and block bootstrap. Finally, we conclude with an illustrative data application.     

Valid Resampling of Higher-Order Statistics Using the Linear Process Bootstrap and Autoregressive Sieve Bootstrap

We show that the linear process bootstrap (LPB) and the autoregressive sieve bootstrap (AR sieve) are, in general, not valid for statistics whose large-sample distribution depends on moments of order higher than two, irrespective of whether the data come from a linear time series or not. Inspired by the block-of-blocks bootstrap, we circumvent this non-validity by applying the LPB and AR sieve to suitably blocked data and not to the original data itself. In a simulation study, we compare the LPB, AR sieve, and moving block bootstrap applied directly and to blocked data.     

An L 2 Comparison Between the Bootstrap and the Empirical Edgeworth Expansion

The Bootstrap estimate for studentized statistics is more accurate than both the normal approximation and the two-term empirical Edgeworth expansion. In this article, it will be shown that the three-term empirical Edgeworth expansion for studentized statistics compares well with the bootstrap. It is also shown that the three-term Edgeworth expansion is superior to the bootstrap in some cases, using more efficient estimators than sample moments in the Edgeworth expansion, such as using maximum likelihood estimators in the one-parameter exponential family.     

Pivotal Quantities Based on Sequential Data: A Bootstrap Approach

A bootstrap algorithm is provided for obtaining a confidence interval for the mean of a probability distribution when sequential data are considered. For this kind of data the empirical distribution can be biased but its bias is bounded by the coefficient of variation of the stopping rule associated with the sequential procedure. When using this distribution for resampling the validity of the bootstrap approach is established by means of a series expansion of the corresponding pivotal quantity. A simulation study is carried out using Wang and Tsiatis type tests and considering the normal and exponential distributions to generate the data. This study confirms that for moderate coefficients of variation of the stopping rule, the bootstrap method allows adequate confidence intervals for the parameters to be obtained, whichever is the distribution of data.     

Moment density estimation for positive random variables

An unknown moment-determinate cumulative distribution function or its density function can be recovered from corresponding moments and estimated from the empirical moments. This method of estimating an unknown density is natural in certain inverse estimation models like multiplicative censoring or biased sampling when the moments of unobserved distribution can be estimated via the transformed moments of the observed distribution. In this paper, we introduce a new nonparametric estimator of a probability density function defined on the positive real line, motivated by the above. Some fundamental properties of proposed estimator are studied. The comparison with traditional kernel density estimator is discussed.     

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.     

Comparison of two methods in estimating standard error of the method of simulated moments estimators for generalized linear mixed models

ABSTRACT

The likelihood of a generalized linear mixed model (GLMM) often involves high-dimensional integrals, which in general cannot be computed explicitly. When direct computation is not available, method of simulated moments (MSM) is a fairly simple way to estimate the parameters of interest. In this research, we compared parametric bootstrap (PB) and nonparametric bootstrap methods (NPB) in estimating the standard errors of MSM estimators for GLMM. Simulation results show that when the group size is large, the PB and NPB perform similarly; when group size is medium, NPB performs better than PB in estimating standard errors of the mean.     

Improving the finite sample performance of autoregression estimators in dynamic factor models: A bootstrap approach

We investigate the finite sample properties of the estimator of a persistence parameter of an unobservable common factor when the factor is estimated by the principal components method. When the number of cross-sectional observations is not sufficiently large, relative to the number of time series observations, the autoregressive coefficient estimator of a positively autocorrelated factor is biased downward, and the bias becomes larger for a more persistent factor. Based on theoretical and simulation analyses, we show that bootstrap procedures are effective in reducing the bias, and bootstrap confidence intervals outperform naive asymptotic confidence intervals in terms of the coverage probability.     

Evaluation of methods for interval estimation of model outputs, with application to survival models

When a published statistical model is also distributed as computer software, it will usually be desirable to present the outputs as interval, as well as point, estimates. The present paper compares three methods for approximate interval estimation about a model output, for use when the model form does not permit an exact interval estimate. The methods considered are first-order asymptotics, using second derivatives of the log-likelihood to estimate variance information; higher-order asymptotics based on the signed-root transformation; and the non-parametric bootstrap. The signed-root method is Bayesian, and uses an approximation for posterior moments that has not previously been tested in a real-world application. Use of the three methods is illustrated with reference to a software project arising in medical decision-making, the UKPDS Risk Engine. Intervals from the first-order and signed-root methods are near- identical, and typically 1% wider to 7% narrower than those from the non-parametric bootstrap. The asymptotic methods are markedly faster than the bootstrap method.     

Asymptotic expansions of moments and cumulants

This paper shows how procedures for computing moments and cumulants may themselves be computed from a few elementary identities.Many parameters, such as variance, may be expressed or approximated as linear combinations of products of expectations. The estimates of such parameters may be expressed as the same linear combinations of products of averages. The moments and cumulants of such estimates may be computed in a straightforward way if the terms of the estimates, moments and cumulants are represented as lists and the expectation operation defined as a transformation of lists. Vector space considerations lead to a unique representation of terms and hence to a simplification of results. Basic identities relating variables and their expectations induce transformations of lists, which transformations may be computed from the identities. In this way procedures for complex calculations are computed from basic identities.The procedures permit the calculation of results which would otherwise involve complementary set partitions, k-statistics, and pattern functions. The examples include the calculation of unbiased estimates of cumulants, of cumulants of these, and of moments of bootstrap estimates.     

Intentionally biased bootstrap methods

A class of weighted bootstrap techniques, called biased bootstrap or b-bootstrap methods, is introduced. It is motivated by the need to adjust empirical methods, such as the 'uniform' bootstrap, in a surgical way to alter some of their features while leaving others unchanged. Depending on the nature of the adjustment, the b-bootstrap can be used to reduce bias, or to reduce variance or to render some characteristic equal to a predetermined quantity. Examples of the last application include a b-bootstrap approach to hypothesis testing in nonparametric contexts, where the b-bootstrap enables simulation 'under the null hypothesis', even when the hypothesis is false, and a b-bootstrap competitor to Tibshirani's variance stabilization method. An example of the bias reduction application is adjustment of Nadaraya–Watson kernel estimators to make them competitive with local linear smoothing. Other applications include density estimation under constraints, outlier trimming, sensitivity analysis, skewness or kurtosis reduction and shrinkage.     

Estimating general variable acceptance sampling plans by bootstrap methods

We consider variable acceptance sampling plans that control the lot or process fraction defective, where a specification limit defines acceptable quality. The problem is to find a sampling plan that fulfils some conditions, usually on the operation characteristic. Its calculation heavily depends on distributional properties that, in practice, might be doubtful. If prior data are already available, we propose to estimate the sampling plan by means of bootstrap methods. The bias and standard error of the estimated plan can be assessed easily by Monte Carlo approximation to the respective bootstrap moments. This resampling approach does not require strong assumptions and, furthermore, is a flexible method that can be extended to any statistic that might be informative for the fraction defective in a lot.     

On the heterogeneity of fecundability

We propose to use a general mixing distribution in modeling the heterogeneity of the fecundability of couples. We introduce a sequence of parameters called canonical moments, which is in one to one correspondence with the moments, to characterize the mixing distribution. By using the bootstrap method, we can estimate the standard errors of our estimates. Our method modifies the usual moment estimates so that the resulting mixing distribution is always supported on [0, 1]. Moreover, the downward bias of the moment estimate of the number of support points would be reduced. Our approach can be used for censored data. The application of our technique in finding the sterile subpopulation is also discussed. The theory is illustrated with several data examples and simulations.     

A note on the scale parameter of the dirichlet process

This paper gives an interpretation for the scale parameter of a Dirichlet process when the aim is to estimate a linear functional of an unknown probability distribution. We provide exact first and second posterior moments for such functionals under both informative and noninformative prior specifications. The noninformative case provides a normal approximation to the Bayesian bootstrap.     

Bootstrap-based evaluation of markov-switching time series models

This paper explores the possibility of evaluating the adequacy of Markov-switching time series models by comparing selected functionals (such as the spectral density function and moving empirical moments) obtained from the data with those of the fitted model using a bootstrap algorithm. The proposed model checking procedure is easy to implement and flexible enough to be adapted to a wide variety of models with parameters subject to Markov regime-switching. Examples with real and artificial data illustrate the potential of the methodology.     

Heteroskedasticity-consistent covariance matrix estimation:white's estimator and the bootstrap ∗

This paper considers the issue of estimating the covariance matrix of ordinary least squares estimates in a linear regression model when heteroskedasticity is suspected. We perform Monte Carlo simulation on the White estimator, which is commonly used in.

empirical research, and also on some alternatives based on different bootstrapping schemes. Our results reveal that the White estimator can be considerably biased when the sample size is not very large, that bias correction via bootstrap does not work well, and that the weighted bootstrap estimators tend to display smaller biases than the White estimator and its variants, under both homoskedasticity and heteroskedasticity. Our results also reveal that the presence of (potentially) influential observations in the design matrix plays an important role in the finite-sample performance of the heteroskedasticity-consistent estimators.