Resampling Methods for Time Series Level Crossings

We present an application of subsampling and bootstrap methods for time series to determine the distribution of the estimator of zero crossings. The zero crossings method provides an alternative estimator of the lag-1 autocorrelation coefficient that is reducing the data storage requirements and is more robust with respect to outliers when compared to the classical estimator. The main results here are showing the consistency of subsampling, the consistency of moving block bootstrap, the consistency of non overlapping block bootstrap and the consistency of stationary bootstrap for this estimator. Theorems are formulated for Gaussian processes, elliptically symmetric processes and processes which are transformed Gaussian processes. Theoretical results are illustrated by simulations and practical data analysis. We have also shown that in practice the MBB method behaves better than the subsampling method.     

Block Bootstrapping for Kernel Density Estimators under ψ-Weak Dependence

Block bootstrap methods are applied to kernel-type density estimator and its derivatives for ψ-weakly dependent processes. Nonparametric density estimation is discussed via moving block bootstrap (MBB) and disjoint block bootstrap (DBB). Asymptotic validity is proved for MBB and DBB. A Monte-Carlo experiment compares confidence intervals based on MBB and DBB with an existing method based on normal approximation (NA) in terms of serial correlation, dynamic asymmetry, and conditional heteroscedasticity. The experiment shows that, in cases of substantial serial correlation, MBB and DBB perform better than NA and, in the other cases, MBB and DBB perform as good as NA.     

Block external bootstrap in partially linear models with nonstationary strong mixing error terms

The authors extend the block external bootstrap to partially linear regression models with strongly mixing, nonstationary error terms. In addition to providing an approximate distribution for the semiparametric least square estimator of the parametric component, they propose a consistent estimator of the co‐variance matrix of this estimator.     

Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity

The main contribution of this paper is a proof of the asymptotic validity of the application of the bootstrap to AR(∞) processes with unmodelled conditional heteroskedasticity. We first derive the asymptotic properties of the least-squares estimator of the autoregressive sieve parameters when the data are generated by a stationary linear process with martingale difference errors that are possibly subject to conditional heteroskedasticity of unknown form. These results are then used in establishing that a suitably constructed bootstrap estimator will have the same limit distribution as the least-squares estimator. Our results provide theoretical justification for the use of either the conventional asymptotic approximation based on robust standard errors or the bootstrap approximation of the distribution of autoregressive parameters. A simulation study suggests that the bootstrap approach tends to be more accurate in small samples.     

Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity

The main contribution of this paper is a proof of the asymptotic validity of the application of the bootstrap to AR(∞) processes with unmodelled conditional heteroskedasticity. We first derive the asymptotic properties of the least-squares estimator of the autoregressive sieve parameters when the data are generated by a stationary linear process with martingale difference errors that are possibly subject to conditional heteroskedasticity of unknown form. These results are then used in establishing that a suitably constructed bootstrap estimator will have the same limit distribution as the least-squares estimator. Our results provide theoretical justification for the use of either the conventional asymptotic approximation based on robust standard errors or the bootstrap approximation of the distribution of autoregressive parameters. A simulation study suggests that the bootstrap approach tends to be more accurate in small samples.     

Bootstrap variance and bias estimation in linear models

Let θ be a nonlinear function of the regression parameters and θ be its estimator based on the least-squares method. This paper studies the bootstrap estimators of the variance and bias of θ. The bootstrap estimators are shown to be consistent and asymptotically unbiased under some conditions. Asymptotic orders of the mean squared errors of the bootstrap estimators are also obtained. The bootstrap and the classical linearization method are compared in a simulation study. Discussions about when to use the bootstrap are given.     

On the estimation of non linear functions in stochastic volatility models

Abstract

This paper focuses on the inference of suitable generally non linear functions in stochastic volatility models. In this context, in order to estimate the variance of the proposed estimators, a moving block bootstrap (MBB) approach is suggested and discussed. Under mild assumptions, we show that the MBB procedure is weakly consistent. Moreover, a methodology to choose the optimal length block in the MBB is proposed. Some examples and simulations on the model are also made to show the performance of the proposed procedure.     

Analysis of tidal data via the blockwise bootstrap

We analyze tidal data from Port Mansfield, TX, using Kunsch's blockwise bootstrap in the regression setting. In particular, we estimate the variability of parameter estimates in a harmonic analysis via block subsampling of residuals from a least-squares fit. We see that naive least-squares variance estimates can be either too large or too small, depending on the strength of correlation and the design matrix. We argue that the block bootstrap is a simple, omnibus method of accounting for correlation in a regression model with correlated errors.     

A modified bootstrap procedure for cluster sampling variance estimation of species richness

Variance estimators for probability sample-based predictions of species richness (S) are typically conditional on the sample (expected variance). In practical applications, sample sizes are typically small, and the variance of input parameters to a richness estimator should not be ignored. We propose a modified bootstrap variance estimator that attempts to capture the sampling variance by generating B replications of the richness prediction from stochastically resampled data of species incidence. The variance estimator is demonstrated for the observed richness (SO), five richness estimators, and with simulated cluster sampling (without replacement) in 11 finite populations of forest tree species. A key feature of the bootstrap procedure is a probabilistic augmentation of a species incidence matrix by the number of species expected to be ‘lost’ in a conventional bootstrap resampling scheme. In Monte-Carlo (MC) simulations, the modified bootstrap procedure performed well in terms of tracking the average MC estimates of richness and standard errors. Bootstrap-based estimates of standard errors were as a rule conservative. Extensions to other sampling designs, estimators of species richness and diversity, and estimates of change are possible.     

On preliminary test and shrinkage estimation in linear models with long-memory errors

This paper discusses the problem of estimating a subset of parameters when the complementary subset is possibly redundant, in a linear regression model when the errors are generated from a long-memory process. Such a model arises due to the overmodelling of a situation involving long-memory data. Along with the classical least-squares estimator and restricted least-squares estimator, preliminary test least-squares estimator and shrinkage least-squares estimator are investigated in an asymptotic set-up and their relative performances are studied under contiguous alternatives. The contiguous alternatives under such dependence are fundamentally different from those under the independent errors case.     

A nonparametric estimator of the number of classes based on a stratified random sample

Under stratified random sampling, we develop a k^th-order bootstrap bias-corrected estimator of the number of classes θ which exist in a study region. This research extends Smith and van Belle’s (1984) first-order bootstrap bias-corrected estimator under simple random sampling. Our estimator has applicability for many settings including: estimating the number of animals when there are stratified capture periods, estimating the number of species based on stratified random sampling of subunits (say, quadrats) from the region, and estimating the number of errors/defects in a product based on observations from two or more types of inspectors. When the differences between the strata are large, utilizing stratified random sampling and our estimator often results in superior performance versus the use of simple random sampling and its bootstrap or jackknife [Burnham and Overton (1978)] estimator. The superior performance is often associated with more observed classes, and we provide insights into optimal designation of the strata and optimal allocation of sample sectors to strata.     

Bootstrap for finite populations

The bootstrap method is compared with the classical (linearization) and jackknife procedures for estimating the mean square errors (MSEs) of the ratio estimator and the combined ratio estimator. The initial samples are considered to be selected without replacement, and different procedures for selecting the bootstrap samples with or without replacement from them are examined. The biases, stabilities, coverage probabilities and confidence widths of all the procedures are compared.     

On intra-class correlation coefficient estimation

Consider the problem of estimating the intra-class correlation coefficient of a symmetric normal distribution. In a recent article (Pal and Lim (1999)) it has been shown that the three popular estimators, namely—the maximum likelihood estimator (MLE), the method of moments estimator (MME) and the unique minimum variance unbiased estimator (UMVUE), are second order admissible under the squared error loss function. In this paper we study the performance of the above mentioned estimators in terms of Pitman Nearness Criterion (PNC) as well as Stochastic Domination Criterion (SDC). We then apply the aforementioned estimators to two real life data sets with moderate to large sample sizes, and bootstrap bias as well as mean squared errors are computed to compare the estimators. In terms of overall performance the MME seems most appealing among the three estimators considered here and this is the main contribution of our paper. Formerly University of Southewestern Louisisna     

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.     

Series Estimation in Partially Linear In‐Slide Regression Models

Abstract. The partially linear in‐slide model (PLIM) is a useful tool to make econometric analyses and to normalize microarray data. In this article, by using series approximations and a least squares procedure, we propose a semiparametric least squares estimator (SLSE) for the parametric component and a series estimator for the non‐parametric component. Under weaker conditions than those imposed in the literature, we show that the SLSE is asymptotically normal and that the series estimator attains the optimal convergence rate of non‐parametric regression. We also investigate the estimating problem of the error variance. In addition, we propose a wild block bootstrap‐based test for the form of the non‐parametric component. Some simulation studies are conducted to illustrate the finite sample performance of the proposed procedure. An example of application on a set of economical data is also illustrated.     

The total bootstrap median: a robust and efficient estimator of location and scale for small samples

We propose the total bootstrap median (TBM) as a robust and efficient estimator of location and scale for small samples. We demonstrate its performance by estimating the mean and variance of a variety of distributions. We also show that, if the underlying distribution is unknown and there is either no contamination or low to moderate contamination, the TBM provides a better estimate of the mean, in mean square terms, than the sample mean or the sample median. In addition, the TBM is a better estimator of the variance of the underlying distribution than the sample variance or the square of the bias-corrected median absolute deviation from the median estimator. We also show that the TBM is an explicit L-estimator, which allows a direct study of its properties.     

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.     

Variance Estimation under Two‐Phase Sampling

We consider the variance estimation of the weighted likelihood estimator (WLE) under two‐phase stratified sampling without replacement. Asymptotic variance of the WLE in many semiparametric models contains unknown functions or does not have a closed form. The standard method of the inverse probability weighted (IPW) sample variances of an estimated influence function is then not available in these models. To address this issue, we develop the variance estimation procedure for the WLE in a general semiparametric model. The phase I variance is estimated by taking a numerical derivative of the IPW log likelihood. The phase II variance is estimated based on the bootstrap for a stratified sample in a finite population. Despite a theoretical difficulty of dependent observations due to sampling without replacement, we establish the (bootstrap) consistency of our estimators. Finite sample properties of our method are illustrated in a simulation study.     

Nonlinear IV panel unit root testing under structural breaks in the error variance

The paper examines the behavior of a generalized version of the nonlinear IV unit root test proposed by Chang (2002) when the series’ errors exhibit nonstationary volatility. The leading case of such nonstationary volatility concerns structural breaks in the error variance. We show that the generalized test is not robust to variance changes in general, and illustrate the extent of the resulting size distortions in finite samples. More importantly, we show that pivotality is recovered when using Eicker-White heteroskedasticity-consistent standard errors. This contrasts with the case of Dickey-Fuller unit root tests, for which Eicker-White standard errors do not produce robustness and thus require computationally costly corrections such as the (wild) bootstrap or estimation of the so-called variance profile. The pivotal versions of the generalized IV tests – with or without the correct standard errors – do however have no power in $1/T$ -neighbourhoods of the null. We also study the validity of panel versions of the tests considered here.     

The three-fold nested random effects model

Estimation of the variance components and the mean of the balanced and unbalanced threefold nested design is considered. The relative merits of the following procedures are evaluated: Analysis of variance (ANOVA), maximum likelihood (ML), restricted maximum likelihood (REML), and minimum variance quadratic unbiased estimator (MIVQUE). A new procedure called the weighted analysis of means (WAM) estimator which utilizes prior information on the variance components is proposed. It is found to have optimum properties similar to the REML and MIVQUE, and it is also computationally simpler. For the mean, the overall sample average, grand mean, unweighted mean, and generalized least-squares (GLS) estimator with its weights obtained from the above estimators for the variance components are considered. Comparisons of the above procedures for the variance components and the mean are made from exact expressions for the biases and mean square errors (MSEs) of the estimators and from empirical investigations.