Comparing three bootstrap methods for survey data

Various bootstrap methods for variance estimation and confidence intervals in complex survey data, where sampling is done without replacement, have been proposed in the literature. The oldest, and perhaps the most intuitively appealing, is the without-replacement bootstrap (BWO) method proposed by Gross (1980). Unfortunately, the BWO method is only applicable to very simple sampling situations. We first introduce extensions of the BWO method to more complex sampling designs. The performance of the BWO and two other bootstrap methods, the rescaling bootstrap (Rao and Wu 1988) and the mirror-match bootstrap (Sitter 1992), are then compared through a simulation study. Together these three methods encompass the various bootstrap proposals.     

Empirical Likelihood Inference for the Rao‐Hartley‐Cochran Sampling Design

The Hartley‐Rao‐Cochran sampling design is an unequal probability sampling design which can be used to select samples from finite populations. We propose to adjust the empirical likelihood approach for the Hartley‐Rao‐Cochran sampling design. The approach proposed intrinsically incorporates sampling weights, auxiliary information and allows for large sampling fractions. It can be used to construct confidence intervals. In a simulation study, we show that the coverage may be better for the empirical likelihood confidence interval than for standard confidence intervals based on variance estimates. The approach proposed is simple to implement and less computer intensive than bootstrap. The confidence interval proposed does not rely on re‐sampling, linearization, variance estimation, design‐effects or joint inclusion probabilities.     

Weighted empirical likelihood inference for multiple samples

We propose a weighted empirical likelihood approach to inference with multiple samples, including stratified sampling, the estimation of a common mean using several independent and non-homogeneous samples and inference on a particular population using other related samples. The weighting scheme and the basic result are motivated and established under stratified sampling. We show that the proposed method can ideally be applied to the common mean problem and problems with related samples. The proposed weighted approach not only provides a unified framework for inference with multiple samples, including two-sample problems, but also facilitates asymptotic derivations and computational methods. A bootstrap procedure is also proposed in conjunction with the weighted approach to provide better coverage probabilities for the weighted empirical likelihood ratio confidence intervals. Simulation studies show that the weighted empirical likelihood confidence intervals perform better than existing ones.     

Empirical likelihood confidence intervals for the mean of a population containing many zero values

If a population contains many zero values and the sample size is not very large, the traditional normal approximation‐based confidence intervals for the population mean may have poor coverage probabilities. This problem is substantially reduced by constructing parametric likelihood ratio intervals when an appropriate mixture model can be found. In the context of survey sampling, however, there is a general preference for making minimal assumptions about the population under study. The authors have therefore investigated the coverage properties of nonparametric empirical likelihood confidence intervals for the population mean. They show that under a variety of hypothetical populations, these intervals often outperformed parametric likelihood intervals by having more balanced coverage rates and larger lower bounds. The authors illustrate their methodology using data from the Canadian Labour Force Survey for the year 2000.     

Stationary bootstrapping for realized covariations of high frequency financial data

This paper studies the stationary bootstrap applicability for realized covariations of high frequency asynchronous financial data. The stationary bootstrap method, which is characterized by a block-bootstrap with random block length, is applied to estimate the integrated covariations. The bootstrap realized covariance, bootstrap realized regression coefficient and bootstrap realized correlation coefficient are proposed, and the validity of the stationary bootstrapping for them is established both for large sample and for finite sample. Consistencies of bootstrap distributions are established, which provide us valid stationary bootstrap confidence intervals. The bootstrap confidence intervals do not require a consistent estimator of a nuisance parameter arising from nonsynchronous unequally spaced sampling while those based on a normal asymptotic theory require a consistent estimator. A Monte-Carlo comparison reveals that the proposed stationary bootstrap confidence intervals have better coverage probabilities than those based on normal approximation.     

Small sample performance of jackknife confidence intervals for the james-stein estimator

The primary goal of this paper is to examine the small sample coverage probability and size of jackknife confidence intervals centered at a Stein-rule estimator. A Monte Carlo experiment is used to explore the coverage probabilities and lengths of nominal 90% and 95% delete-one and infinitesimal jackknife confidence intervals centered at the Stein-rule estimator; these are compared to those obtained using a bootstrap procedure.     

Uncertainty estimation in heterogeneous capture–recapture count data

The Conway–Maxwell–Poisson estimator is considered in this paper as the population size estimator. The benefit of using the Conway–Maxwell–Poisson distribution is that it includes the Bernoulli, the Geometric and the Poisson distributions as special cases and, furthermore, allows for heterogeneity. Little emphasis is often placed on the variability associated with the population size estimate. This paper provides a deep and extensive comparison of bootstrap methods in the capture–recapture setting. It deals with the classical bootstrap approach using the true population size, the true bootstrap, and the classical bootstrap using the observed sample size, the reduced bootstrap. Furthermore, the imputed bootstrap, as well as approximating forms in terms of standard errors and confidence intervals for the population size, under the Conway–Maxwell–Poisson distribution, have been investigated and discussed. These methods are illustrated in a simulation study and in benchmark real data examples.     

Bootstrap confidence intervals of CNpk for inverse Rayleigh and log-logistic distributions

In this article bootstrap confidence intervals of process capability index as suggested by Chen and Pearn [An application of non-normal process capability indices. Qual Reliab Eng Int. 1997;13:355–360] are studied through simulation when the underlying distributions are inverse Rayleigh and log-logistic distributions. The well-known maximum likelihood estimator is used to estimate the parameter. The bootstrap confidence intervals considered in this paper consists of various confidence intervals. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Application examples on two distributions for process capability indices are provided for practical use.     

Pseudo‐empirical likelihood ratio confidence intervals for complex surveys

The authors show how an adjusted pseudo‐empirical likelihood ratio statistic that is asymptotically distributed as a chi‐square random variable can be used to construct confidence intervals for a finite population mean or a finite population distribution function from complex survey samples. They consider both non‐stratified and stratified sampling designs, with or without auxiliary information. They examine the behaviour of estimates of the mean and the distribution function at specific points using simulations calling on the Rao‐Sampford method of unequal probability sampling without replacement. They conclude that the pseudo‐empirical likelihood ratio confidence intervals are superior to those based on the normal approximation, whether in terms of coverage probability, tail error rates or average length of the intervals.     

Transformed Logit Confidence Intervals for Small Populations in Single Capture–Recapture Estimation

The good performance of logit confidence intervals for the odds ratio with small samples is well known. This is true unless the actual odds ratio is very large. In single capture–recapture estimation the odds ratio is equal to 1 because of the assumption of independence of the samples. Consequently, a transformation of the logit confidence intervals for the odds ratio is proposed in order to estimate the size of a closed population under single capture–recapture estimation. It is found that the transformed logit interval, after adding .5 to each observed count before computation, has actual coverage probabilities near to the nominal level even for small populations and even for capture probabilities near to 0 or 1, which is not guaranteed for the other capture–recapture confidence intervals proposed in statistical literature. Thus, given that the .5 transformed logit interval is very simple to compute and has a good performance, it is appropriate to be implemented by most users of the single capture–recapture method.     

Estimators in capture–recapture studies with two sources

This paper investigates the applications of capture–recapture methods to human populations. Capture–recapture methods are commonly used in estimating the size of wildlife populations but can also be used in epidemiology and social sciences, for estimating prevalence of a particular disease or the size of the homeless population in a certain area. Here we focus on estimating the prevalence of infectious diseases. Several estimators of population size are considered: the Lincoln–Petersen estimator and its modified version, the Chapman estimator, Chao’s lower bound estimator, the Zelterman’s estimator, McKendrick’s moment estimator and the maximum likelihood estimator. In order to evaluate these estimators, they are applied to real, three-source, capture-recapture data. By conditioning on each of the sources of three source data, we have been able to compare the estimators with the true value that they are estimating. The Chapman and Chao estimators were compared in terms of their relative bias. A variance formula derived through conditioning is suggested for Chao’s estimator, and normal 95% confidence intervals are calculated for this and the Chapman estimator. We then compare the coverage of the respective confidence intervals. Furthermore, a simulation study is included to compare Chao’s and Chapman’s estimator. Results indicate that Chao’s estimator is less biased than Chapman’s estimator unless both sources are independent. Chao’s estimator has also the smaller mean squared error. Finally, the implications and limitations of the above methods are discussed, with suggestions for further development. We are grateful to the Medical Research Council for supporting this work.     

Weighted analyses for cohort sampling designs

Weighted analysis methods are considered for cohort sampling designs that allow subsampling of both cases and non-cases, but with cases generally sampled more intensively. The methods fit into the general framework for the analysis of survey sampling designs considered by Lin (Biometrika 87:37–47, 2000). Details are given for applying the general methodology in this setting. In addition to considering proportional hazards regression, methods for evaluating the representativeness of the sample and for estimating event-free probabilities are given. In a small simulation study, the one-sample cumulative hazard estimator and its variance estimator were found to be nearly unbiased, but the true coverage probabilities of confidence intervals computed from these sometimes deviated significantly from the nominal levels. Methods for cross-validation and for bootstrap resampling, which take into account the dependencies in the sample, are also considered. An erratum to this article can be found at     

Rao and Wu's re-scaling bootstrap modified to achieve extended coverages

Horvitz and Thompson's (HT) [1952. A generalization of sampling without replacement from a finite universe. J. Amer. Statist. Assoc. 47, 663–685] well-known unbiased estimator for a finite population total admits an unbiased estimator for its variance as given by [Yates and Grundy, 1953. Selection without replacement from within strata with probability proportional to size. J. Roy. Statist. Soc. B 15, 253–261], provided the parent sampling design involves a constant number of distinct units in every sample to be chosen. If the design, in addition, ensures uniform non-negativity of this variance estimator, Rao and Wu [1988. Resampling inference with complex survey data. J. Amer. Statist. Assoc. 83, 231–241] have given their re-scaling bootstrap technique to construct confidence interval and to estimate mean square error for non-linear functions of finite population totals of several real variables. Horvitz and Thompson's estimators (HTE) are used to estimate the finite population totals. Since they need to equate the bootstrap variance of the bootstrap estimator to the Yates and Grundy's estimator (YGE) for the variance of the HTE in case of a single variable, i.e., in the linear case the YG variance estimator is required to be positive for the sample usually drawn.     

On the integrated maximum likelihood estimators for a closed population capture–recapture model with unequal capture probabilities

Nuisance parameter elimination is a central problem in capture–recapture modelling. In this paper, we consider a closed population capture–recapture model which assumes the capture probabilities varies only with the sampling occasions. In this model, the capture probabilities are regarded as nuisance parameters and the unknown number of individuals is the parameter of interest. In order to eliminate the nuisance parameters, the likelihood function is integrated with respect to a weight function (uniform and Jeffrey's) of the nuisance parameters resulting in an integrated likelihood function depending only on the population size. For these integrated likelihood functions, analytical expressions for the maximum likelihood estimates are obtained and it is proved that they are always finite and unique. Variance estimates of the proposed estimators are obtained via a parametric bootstrap resampling procedure. The proposed methods are illustrated on a real data set and their frequentist properties are assessed by means of a simulation study.     

ON THE ANALYSIS AND APPLICATION OF MEASURES OF LINKAGE DISEQUILIBRIUM

The maximum likelihood, jackknife and bootstrap estimators of linkage disequilibrium, a measure of association in population genetics, are derived and compared. It is found that for point estimation, the resampling methods generate almost identical mean square errors. The maximum likelihood estimator could have bigger or smaller mean square errors depending on the parameters of the underlying population. However the bootstrap confidence interval is superior to the other two as the length of the intervals is shorter or the probability that the 95% confidence intervals include the true parameter is closer to 0.95. Although the standardised measure of linkage disequilibrium has a range from -1 to 1 regardless of marginal frequencies, it is shown that the distribution of this standardised measure is still not allele frequency independent under the multinomial sampling scheme.     

Empirical likelihood inference for a common mean in the presence of heteroscedasticity

The authors develop empirical likelihood (EL) based methods of inference for a common mean using data from several independent but nonhomogeneous populations. For point estimation, they propose a maximum empirical likelihood (MEL) estimator and show that it is n‐consistent and asymptotically optimal. For confidence intervals, they consider two EL based methods and show that both intervals have approximately correct coverage probabilities under large samples. Finite‐sample performances of the MEL estimator and the EL based confidence intervals are evaluated through a simulation study. The results indicate that overall the MEL estimator and the weighted EL confidence interval are superior alternatives to the existing methods.     

Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions

One of the indicators for evaluating the capability of a process is the process capability index. In this article, bootstrap confidence intervals of the generalized process capability index (GPCI) proposed by Maiti et al. are studied through simulation, when the underlying distributions are Lindley and Power Lindley distributions. The maximum likelihood method is used to estimate the parameters of the models. Three bootstrap confidence intervals namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile bootstrap (BCPB) are considered for obtaining confidence intervals of GPCI. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average width of the bootstrap confidence intervals. Simulation results show that the estimated coverage probabilities of the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval get closer to the nominal confidence level than those of the standard bootstrap confidence interval. Finally, three real datasets are analyzed for illustrative purposes.     

Performance of Interval Estimators for the Inverse Hypergeometric Distribution

The inverse hypergeometric distribution is of interest in applications of inverse sampling without replacement from a finite population where a binary observation is made on each sampling unit. Thus, sampling is performed by randomly choosing units sequentially one at a time until a specified number of one of the two types is selected for the sample. Assuming the total number of units in the population is known but the number of each type is not, we consider the problem of estimating this parameter. We use the Delta method to develop approximations for the variance of three parameter estimators. We then propose three large sample confidence intervals for the parameter. Based on these results, we selected a sampling of parameter values for the inverse hypergeometric distribution to empirically investigate performance of these estimators. We evaluate their performance in terms of expected probability of parameter coverage and confidence interval length calculated as means of possible outcomes weighted by the appropriate outcome probabilities for each parameter value considered. The unbiased estimator of the parameter is the preferred estimator relative to the maximum likelihood estimator and an estimator based on a negative binomial approximation, as evidenced by empirical estimates of closeness to the true parameter value. Confidence intervals based on the unbiased estimator tend to be shorter than the two competitors because of its relatively small variance but at a slight cost in terms of coverage probability.     

Approximate confidence interval construction for risk difference under inverse sampling

For studies with dichotomous outcomes, inverse sampling (also known as negative binomial sampling) is often used when the subjects arrive sequentially, when the underlying response of interest is acute, and/or when the maximum likelihood estimators of some epidemiologic indices are undefined. Although exact unconditional inference has been shown to be appealing, its applicability and popularity is severely hindered by the notorious conservativeness due to the adoption of the maximization principle and by the tedious computing time due to the involvement of infinite summation. In this article, we demonstrate how these obstacles can be overcome by the application of the constrained maximum likelihood estimation and truncated approximation. The present work is motivated by confidence interval construction for the risk difference under inverse sampling. Wald-type and score-type confidence intervals based on inverting two one-sided and one two-sided tests are considered. Monte Carlo simulations are conducted to evaluate the performance of these confidence intervals with respect to empirical coverage probability, empirical confidence width, and empirical left and right non-coverage probabilities. Two examples from a maternal congenital heart disease study and a drug comparison study are used to demonstrate the proposed methodologies.