On an Asymptotic Theory of Conditional and Unconditional Coverage Probabilities of Empirical Bayes Confidence Intervals

Empirical Bayes (EB) methodology is now widely used in statistics. However, construction of EB confidence intervals is still very limited. Following Cox (1975 ), Hill (1990 ) and Carlin & Gelfand (1990 , 1991 ), we consider EB confidence intervals, which are adjusted so that the actual coverage probabilities asymptotically meet the target coverage probabilities up to the second order. We consider both unconditional and conditional coverage, conditioning being done with respect to an ancillary statistic.     

What Do Interpolated Nonparametric Confidence Intervals for Population Quantiles Guarantee?

The interval between two prespecified order statistics of a sample provides a distribution-free confidence interval for a population quantile. However, due to discreteness, only a small set of exact coverage probabilities is available. Interpolated confidence intervals are designed to expand the set of available coverage probabilities. However, we show here that the infimum of the coverage probability for an interpolated confidence interval is either the coverage probability for the inner interval or the coverage probability obtained by removing the more likely of the two extreme subintervals from the outer interval. Thus, without additional assumptions, interpolated intervals do not expand the set of available guaranteed coverage probabilities.     

Bias corrected empirical Bayes confidence intervals for the selected mean

Empirical Bayes (EB) methods are very useful for post selection inference. Following Datta et al. (2002 Datta, G. S., M. Ghosh, D. D. Smith, and P. Lahiri. 2002. On an asymptotic theory of conditional and unconditional coverage probabilities of empirical Bayes confidence intervals. Scandinavian Journal of Statistics 29:139–52.[Crossref], [Web of Science ®] , [Google Scholar]), we construct EB confidence intervals for the selected population mean. The EB intervals are adjusted to achieve the target coverage probabilities asymptotically up to the second order. Both unconditional coverage probabilities of EB intervals and corresponding probabilities conditional on ancillary statistics are found.     

Corrected empirical Bayes confidence intervals in nested error regression models

In the small area estimation, the empirical best linear unbiased predictor (EBLUP) or the empirical Bayes estimator (EB) in the linear mixed model is recognized to be useful because it gives a stable and reliable estimate for a mean of a small area. In practical situations where EBLUP is applied to real data, it is important to evaluate how much EBLUP is reliable. One method for the purpose is to construct a confidence interval based on EBLUP. In this paper, we obtain an asymptotically corrected empirical Bayes confidence interval in a nested error regression model with unbalanced sample sizes and unknown components of variance. The coverage probability is shown to satisfy the confidence level in the second-order asymptotics. It is numerically revealed that the corrected confidence interval is superior to the conventional confidence interval based on the sample mean in terms of the coverage probability and the expected width of the interval. Finally, it is applied to the posted land price data in Tokyo and the neighboring prefecture.     

Properties of simultaneous confidence intervals for multinomial proportions

Inversion of Pearson's chi-square statistic yields a confidence ellipsoid that can be used for simultaneous inference concerning multinomial proportions. Because the ellipsoid is difficult to interpret, methods of simultaneous confidence interval construction have been proposed by Quesenberry and hurst,goodman,fitzpatrick and scott and sison and glaz . Based on simulation results, we discuss the performance of these methods in terms of empirical coverage probabilities and enclosed volume. None of the methods is uniformly better than all others, but the Goodman intervals control the empirical coverage probability with smaller volume than other methods when the sample size supports the large sample theory. If the expected cell counts are small and nearly equal across cells, we recommend the sison and glaz intervals.     

Confidence intervals for coefficients of variation in two-parameter exponential distributions

This article examines confidence intervals for the single coefficient of variation and the difference of coefficients of variation in the two-parameter exponential distributions, using the method of variance of estimates recovery (MOVER), the generalized confidence interval (GCI), and the asymptotic confidence interval (ACI). In simulation, the results indicate that coverage probabilities of the GCI maintain the nominal level in general. The MOVER performs well in terms of coverage probability when data only consist of positive values, but it has wider expected length. The coverage probabilities of the ACI satisfy the target for large sample sizes. We also illustrate our confidence intervals using a real-world example in the area of medical science.     

Interpreting statistical evidence by using imperfect models: robust adjusted likelihood functions

Summary. The strength of statistical evidence is measured by the likelihood ratio. Two key performance properties of this measure are the probability of observing strong misleading evidence and the probability of observing weak evidence. For the likelihood function associated with a parametric statistical model, these probabilities have a simple large sample structure when the model is correct. Here we examine how that structure changes when the model fails. This leads to criteria for determining whether a given likelihood function is robust (continuing to perform satisfactorily when the model fails), and to a simple technique for adjusting both likelihoods and profile likelihoods to make them robust. We prove that the expected information in the robust adjusted likelihood cannot exceed the expected information in the likelihood function from a true model. We note that the robust adjusted likelihood is asymptotically fully efficient when the working model is correct, and we show that in some important examples this efficiency is retained even when the working model fails. In such cases the Bayes posterior probability distribution based on the adjusted likelihood is robust, remaining correct asymptotically even when the model for the observable random variable does not include the true distribution. Finally we note a link to standard frequentist methodology—in large samples the adjusted likelihood functions provide robust likelihood-based confidence intervals.     

Confidence and prediction intervals based on interpolated records

In several statistical problems, nonparametric confidence intervals for population quantiles can be constructed and their coverage probabilities can be computed exactly, but cannot in general be rendered equal to a pre-determined level. The same difficulty arises for coverage probabilities of nonparametric prediction intervals for future observations. One solution to this difficulty is to interpolate between intervals which have the closest coverage probability from above and below to the pre-determined level. In this paper, confidence intervals for population quantiles are constructed based on interpolated upper and lower records. Subsequently, prediction intervals are obtained for future upper records based on interpolated upper records. Additionally, we derive upper bounds for the coverage error of these confidence and prediction intervals. Finally, our results are applied to some real data sets. Also, a comparison via a simulation study is done with similar classical intervals obtained before.     

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.     

An empirical bayes approach to a variables acceptance sampling plan problem

An empirical Bayes approach to a variables acceptance sampling plan problem is presented and an empirical Bayes rule is developed which is shown to be asymptotically optimal under general conditions. The problem considered is one in which the ratio of the costs of accepting defective items and rejecting non-defective items is specified. Sampling costs are not considered and the size of the sample taken from each lot is fixed and constant. The empirical Bayes estimation of the Bayes rule is shown to require the estimation of a conditional probability. An estimator for conditional probabilities of the form needed is derived and shown to have good asymptotic properties.     

A matching prior for the product of normal means based on the modified profile likelihood

In this paper, we develop a matching prior for the product of means in several normal distributions with unrestricted means and unknown variances. For this problem, properly assigning priors for the product of normal means has been issued because of the presence of nuisance parameters. Matching priors, which are priors matching the posterior probabilities of certain regions with their frequentist coverage probabilities, are commonly used but difficult to derive in this problem. We developed the first order probability matching priors for this problem; however, the developed matching priors are unproper. Thus, we apply an alternative method and derive a matching prior based on a modification of the profile likelihood. Simulation studies show that the derived matching prior performs better than the uniform prior and Jeffreys’ prior in meeting the target coverage probabilities, and meets well the target coverage probabilities even for the small sample sizes. In addition, to evaluate the validity of the proposed matching prior, Bayesian credible interval for the product of normal means using the matching prior is compared to Bayesian credible intervals using the uniform prior and Jeffrey’s prior, and the confidence interval using the method of Yfantis and Flatman.     

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.     

Bayesian inference and prediction of order statistics for a Type-II censored Weibull distribution

This paper describes the Bayesian inference and prediction of the two-parameter Weibull distribution when the data are Type-II censored data. The aim of this paper is twofold. First we consider the Bayesian inference of the unknown parameters under different loss functions. The Bayes estimates cannot be obtained in closed form. We use Gibbs sampling procedure to draw Markov Chain Monte Carlo (MCMC) samples and it has been used to compute the Bayes estimates and also to construct symmetric credible intervals. Further we consider the Bayes prediction of the future order statistics based on the observed sample. We consider the posterior predictive density of the future observations and also construct a predictive interval with a given coverage probability. Monte Carlo simulations are performed to compare different methods and one data analysis is performed for illustration purposes.     

Improved nonparametric tolerance intervals based on interpolated and extrapolated order statistics

The standard approach to construct nonparametric tolerance intervals is to use the appropriate order statistics, provided a minimum sample size requirement is met. However, it is well-known that this traditional approach is conservative with respect to the nominal level. One way to improve the coverage probabilities is to use interpolation. However, the extension to the case of two-sided tolerance intervals, as well as for the case when the minimum sample size requirement is not met, have not been studied. In this paper, an approach using linear interpolation is proposed for improving coverage probabilities for the two-sided setting. In the case when the minimum sample size requirement is not met, coverage probabilities are shown to improve by using linear extrapolation. A discussion about the effect on coverage probabilities and expected lengths when transforming the data is also presented. The applicability of this approach is demonstrated using three real data sets.     

Bootstrap adjustments for empirical bayes interval estimates of small-area proportions

Empirical Bayes approaches have often been applied to the problem of estimating small-area parameters. As a compromise between synthetic and direct survey estimators, an estimator based on an empirical Bayes procedure is not subject to the large bias that is sometimes associated with a synthetic estimator, nor is it as variable as a direct survey estimator. Although the point estimates perform very well, naïve empirical Bayes confidence intervals tend to be too short to attain the desired coverage probability, since they fail to incorporate the uncertainty which results from having to estimate the prior distribution. Several alternative methodologies for interval estimation which correct for the deficiencies associated with the naïve approach have been suggested. Laird and Louis (1987) proposed three types of bootstrap for correcting naïve empirical Bayes confidence intervals. Calling the methodology of Laird and Louis (1987) an unconditional bias-corrected naïve approach, Carlin and Gelfand (1991) suggested a modification to the Type III parametric bootstrap which corrects for bias in the naïve intervals by conditioning on the data. Here we empirically evaluate the Type II and Type III bootstrap proposed by Laird and Louis, as well as the modification suggested by Carlin and Gelfand (1991), with the objective of examining coverage properties of empirical Bayes confidence intervals for small-area proportions.     

Empirical likelihood for generalized linear models with missing responses

The paper uses the empirical likelihood method to study the construction of confidence intervals and regions for regression coefficients and response mean in generalized linear models with missing response. By using the inverse selection probability weighted imputation technique, the proposed empirical likelihood ratios are asymptotically chi-squared. Our approach is to directly calibrate the empirical likelihood ratio, which is called as a bias-correction method. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths/areas of confidence intervals/regions. An example of a real data set is used for illustrating our methods.     

Comparison among non parametric prediction intervals of order statistics

Abstract

In this article, we are interested in conducting a comparison study between different non parametric prediction intervals of order statistics from a future sample based on an observed order statistics. Typically, coverage probabilities of well-known non parametric prediction intervals may not reach the preassigned probability levels. Moreover, prediction intervals for predicting future order statistics are no longer available in some cases. For this, we propose different methods involving random indices and fractional order statistics. In each case, we find the optimal prediction intervals. Numerical computations are presented to assess the performances of the so-obtained intervals. Finally, a real-life data set is presented and analyzed for illustrative purposes.     

Coverage Properties of Confidence Intervals for Generalized Additive Model Components

Abstract. We study the coverage properties of Bayesian confidence intervals for the smooth component functions of generalized additive models (GAMs) represented using any penalized regression spline approach. The intervals are the usual generalization of the intervals first proposed by Wahba and Silverman in 1983 and 1985, respectively, to the GAM component context. We present simulation evidence showing these intervals have close to nominal ‘across‐the‐function’ frequentist coverage probabilities, except when the truth is close to a straight line/plane function. We extend the argument introduced by Nychka in 1988 for univariate smoothing splines to explain these results. The theoretical argument suggests that close to nominal coverage probabilities can be achieved, provided that heavy oversmoothing is avoided, so that the bias is not too large a proportion of the sampling variability. The theoretical results allow us to derive alternative intervals from a purely frequentist point of view, and to explain the impact that the neglect of smoothing parameter variability has on confidence interval performance. They also suggest switching the target of inference for component‐wise intervals away from smooth components in the space of the GAM identifiability constraints.     

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.     

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.