Jackknifing the Mantel–Haenszel Estimator in Sparse Contingency Tables

This article investigates the performance of two jackknife techniques under an asymptotic model in which the number of 2 × 2 tables increases but the possible marginal configurations remain fixed. These approaches are applied to the Mantel–Haenszel estimator, or transformed versions of this estimator, respectively. The resulting jackknife estimators are shown to be consistent for the common odds ratio. Their asymptotic distributions are derived; they can be used for constructing appropriate sparse-data confidence intervals.     

Jackknife inference for heteroscedastic linear regression models

Inference on the regression parameters in a heteroscedastic linear regression model with replication is considered, using either the ordinary least-squares (OLS) or the weighted least-squares (WLS) estimator. A delete-group jackknife method is shown to produce consistent variance estimators irrespective of within-group correlations, unlike the delete-one jackknife variance estimators or those based on the customary δ-method assuming within-group independence. Finite-sample properties of the delete-group variance estimators and associated confidence intervals are also studied through simulation.     

Repeated measurement analysis for nonnormal data in small samples

The small sample performance of Zeger and Liang's extended generalized linear models for the analysis of longitudinal data (Biometrics, 42,121-130,1986) is investigated for correlated gamma data. Results show that the confidence intervals do not provide desirable coverage of the true parameter due to considerably biased point estimates. Improved estimates are proposed using the jackknife procedure. Simulations performed to evaluate the proposed estimates indicate superior properties to the previous estimates.     

Mem squared error estimation in finite populations under nonlinear models

Four estimators of the prediction mean squared error (MSB) of an estimated finite population total for a zero-one characteristic are examined. The characteristic associated with each population unit is modeled as the realization of a Bernoulli random variable whose expected value is a nonlinear function of a parameter vector and a set of known auxiliary variables. To compare the estimators, a simulation study is conducted using a population of hospitals. The MSB estimator Implied by the form of the assumed model underestimates the mean squared error in each of the cases studied and produces confidence lntervals with less than the nominal coverage probabilities. Of the three alternative MSE estimators presented, a linear approximation to the jackknife produces the best results and improves upon the model-specific estimator.     

The Jackknife applied to incomplete blood sampling models

A nonparametric alternative to the extended parametric Satterthwaite method taking into account the correlation between samples obtained from the same animal is presented. Confidence intervals are computed using a modified jackknife standard error. The performance of the method is studied by means of simulation. This nonparametric method has a smaller coverage rate than the parametric one, but still close to the nominal confidence level. In addition, its confidence interval is shorter than that of the parametric method using the generalized Satterthwaite approximation. Copyright © 2003 John Wiley & Sons, Ltd.     

Nonparametric standard errors and confidence intervals

We investigate several nonparametric methods; the bootstrap, the jackknife, the delta method, and other related techniques. The first and simplest goal is the assignment of nonparametric standard errors to a real-valued statistic. More ambitiously, we consider setting nonparametric confidence intervals for a real-valued parameter. Building on the well understood case of confidence intervals for the median, some hopeful evidence is presented that such a theory may be possible.     

Half-sample variance estimation

This paper studies an alternative to the jackknife variance estimator, the half-sample variance estimator. Both theoretical and Monte Carlo comparisons between the half-sample variance estimator and the jackknife variance estimator indicate that the former is better in some situations.     

On using the jackknife to estimate quantile variance

We show that the jackknife technique fails badly when applied to the problem of estimating the variance of a sample quantile. When viewed as a point estimator, the jackknife estimator is known to be inconsistent. We show that the ratio of the jackknife variance estimate to the true variance has an asymptotic Weibull distribution with parameters 1 and 1/2. We also show that if the jackknife variance estimate is used to Studentize the sample quantile, the asymptotic distribution of the resulting Studentized statistic is markedly nonnormal, having infinite mean. This result is in stark contrast with that obtained in simpler problems, such as that of constructing confidence intervals for a mean, where the jackknife-Studentized statistic has an asymptotic standard normal distribution.     

On Nonsmooth Estimating Functions via Jackknife Empirical Likelihood

In many applications, the parameters of interest are estimated by solving non‐smooth estimating functions with U‐statistic structure. Because the asymptotic covariances matrix of the estimator generally involves the underlying density function, resampling methods are often used to bypass the difficulty of non‐parametric density estimation. Despite its simplicity, the resultant‐covariance matrix estimator depends on the nature of resampling, and the method can be time‐consuming when the number of replications is large. Furthermore, the inferences are based on the normal approximation that may not be accurate for practical sample sizes. In this paper, we propose a jackknife empirical likelihood‐based inferential procedure for non‐smooth estimating functions. Standard chi‐square distributions are used to calculate the p‐value and to construct confidence intervals. Extensive simulation studies and two real examples are provided to illustrate its practical utilities.     

Robust likelihood inference for public policy

Issues of public policy are typically decided by non‐specialists who are increasingly informed by statistical methods. In order to be influential, inferential techniques must be widely understood and accepted. This motivates the author to propose likelihood‐based methods that prove relatively insensitive to the choice of underlying distribution because they exploit a peculiarly stable relation between two standard errors and a 95% coverage probability. The author also notes that bootstrap and jackknife estimates of variance can sometimes be strongly biased. In fact, symbolic computations in R suggest that they are reliable only for statistics that are well approximated by averages whose distributions are roughly symmetric. The author thus proposes to transform the classical likelihood ratio into a statistic whose variance can be estimated robustly. He shows that the signed root of the log‐likelihood is well approximated by an average with a roughly symmetric distribution. This leads to Cox‐Tukey intervals for a Student‐like statistic and to simple confidence intervals for most models used in public policy.     

Consistency of least-squares estimator and its jackknife variance estimator in nonlinear models

The purpose of this paper is twofold: (1) We establish the consistency of the least-squares estimator in a nonlinear modely_i = f(x_i,θ) +σ_ie_i where the range of the parameter θ is noncompact, the regression function is unbounded, and the σ_i,'s are not necessarily equal. This extends the results in Jennrich (1969) and Wu (1981). (2) Under the same model, the jackknife estimator of the asymptotic covariance matrix of the least-squares estimator is shown to be consistent, which provides a theoretical justification of the empirical results in Duncan (1978) and the use of the jackknife method in large-sample inferences.     

Delete-m Jackknife for Unequal m

In this paper, the delete-mj jackknife estimator is proposed. This estimator is based on samples obtained from the original sample by successively removing mutually exclusive groups of unequal size. In a Monte Carlo simulation study, a hierarchical linear model was used to evaluate the role of nonnormal residuals and sample size on bias and efficiency of this estimator. It is shown that bias is reduced in exchange for a minor reduction in efficiency. The accompanying jackknife variance estimator even improves on both bias and efficiency, and, moreover, this estimator is mean-squared-error consistent, whereas the maximum likelihood equivalents are not.     

On the bootstrap in cube root asymptotics

The authors study the application of the bootstrap to a class of estimators which converge at a nonstandard rate to a nonstandard distribution. They provide a theoretical framework to study its asymptotic behaviour. A simulation study shows that in the case of an estimator such as Chernoff's estimator of the mode, usually the basic bootstrap confidence intervals drastically undercover while the percentile bootstrap intervals overcover. This is a rare instance where basic and percentile confidence intervals, which have exactly the same length, behave in a very different way. In the case of Chernoff's estimator, if the distribution is symmetric, it is possible to bootstrap from a smooth symmetric estimator of the distribution for which the basic bootstrap confidence intervals will have the claimed coverage probability while the percentile bootstrap interval will have an asymptotic coverage of 1!     

A note on the two-sample mean problem based on jackknife empirical likelihood

In this article, we employ the jackknife empirical likelihood (JEL) method to construct the confidence regions for the difference of the means of two d-dimensional samples. Compared with traditional EL for the two-sample mean problem, JEL is extremely simpler to use in practice and is more effective in computing. Based on the JEL ratio test, a version of Wilks’ theorem is developed. Furthermore, to improve the coverage accuracy of confidence regions, a Bartlett correction is applied. The effectiveness of the proposed method is demonstrated by a simulation study and a real data analysis.     

Better nonparametric bootstrap confidence intervals for the correlation coefficient

We respond to criticism leveled at bootstrap confidence intervals for the correlation coefficient by recent authors by arguing that in the correlation coefficient case, non–standard methods should be employed. We propose two such methods. The first is a bootstrap coverage coorection algorithm using iterated bootstrap techniques (Hall, 1986; Beran, 1987a; Hall and Martin, 1988) applied to ordinary percentile–method intervals (Efron, 1979), giving intervals with high coverage accuracy and stable lengths and endpoints. The simulation study carried out for this method gives results for sample sizes 8, 10, and 12 in three parent populations. The second technique involves the construction of percentile–t bootstrap confidence intervals for a transformed correlation coefficient, followed by an inversion of the transformation, to obtain “transformed percentile–t” intervals for the correlation coefficient. In particular, Fisher's z–transformation is used, and nonparametric delta method and jackknife variance estimates are used to Studentize the transformed correlation coefficient, with the jackknife–Studentized transformed percentile–t interval yielding the better coverage accuracy, in general. Percentile–t intervals constructed without first using the transformation perform very poorly, having large expected lengths and erratically fluctuating endpoints. The simulation study illustrating this technique gives results for sample sizes 10, 15 and 20 in four parent populations. Our techniques provide confidence intervals for the correlation coefficient which have good coverage accuracy (unlike ordinary percentile intervals), and stable lengths and endpoints (unlike ordinary percentile–t intervals).     

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.     

A Note on Simultaneous Confidence Intervals for Ratio of Variances

An explicit formula for confidence intervals for ratios of variances of several populations is presented. The intervals are based on jackknife statistics and the critical point of the studentized range distribution. The asymptotic probability of coverage is not less than the nominal value provided that the distributions of the sampled populations belong to a location-scale family of probabilities with finite fourth moment.     

A comparative study of confidence intervals for negative binomial proportion

In this paper, we investigate four existing and three new confidence interval estimators for the negative binomial proportion (i.e., proportion under inverse/negative binomial sampling). An extensive and systematic comparative study among these confidence interval estimators through Monte Carlo simulations is presented. The performance of these confidence intervals are evaluated in terms of their coverage probabilities and expected interval widths. Our simulation studies suggest that the confidence interval estimator based on saddlepoint approximation is more appealing for large coverage levels (e.g., nominal level≤1% ) whereas the score confidence interval estimator is more desirable for those commonly used coverage levels (e.g., nominal level>1% ). We illustrate these confidence interval construction methods with a real data set from a maternal congenital heart disease study.     

Generalized Jackknife-Based Estimators for Univariate Extreme-Value Modeling

In this article, we revisit the importance of the generalized jackknife in the construction of reliable semi-parametric estimates of some parameters of extreme or even rare events. The generalized jackknife statistic is applied to a minimum-variance reduced-bias estimator of a positive extreme value index—a primary parameter in statistics of extremes. A couple of refinements are proposed and a simulation study shows that these are able to achieve a lower mean square error. A real data illustration is also provided.     

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.