Invariance,Optimality, and a 1-Observation Confidence Interval for a Normal Mean

In a 1965 Decision Theory course at Stanford University, Charles Stein began a digression with “an amusing problem”: is there a proper confidence interval for the mean based on a single observation from a normal distribution with both mean and variance unknown? Stein introduced the interval with endpoints ± c|X| and showed indeed that for c large enough, the minimum coverage probability (over all values for the mean and variance) could be made arbitrarily near one. While the problem and coverage calculation were in the author’s hand-written notes from the course, there was no development of any optimality result for the interval. Here, the Hunt–Stein construction plus analysis based on special features of the problem provides a “minimax” rule in the sense that it minimizes the maximum expected length among all procedures with fixed coverage (or, equivalently, maximizes the minimal coverage among all procedures with a fixed expected length). The minimax rule is a mixture of two confidence procedures that are equivariant under scale and sign changes, and are uniformly better than the classroom example or the natural interval X ± c|X|?.     

Confidence sets and the stein effect

The problem of improving upon the usual set estimator of a multivariate normal mean has only recently seen significant advances. Improved sets that take advantage of the Stein effect have been constructed. It is shown here that the Stein effect is so powerful that one can construct improved confidence sets that can have zero radius on a set of positive probability. Other, somewhat more sensible, sets which attain arbitrarily small radius are also constructed, and it is argued that one way to eliminate unreasonable confidence sets is through a conditional evaluation.     

Estimated confidence under ancillary statistic everywhere-valid constraint

Consider the problem of estimating the coverage function of an usual confidence interval for a randomly chosen linear combination of the elements of the mean vector of a p-dimensional normal distribution. The usual constant coverage probability estimator is shown to be admissible under the ancillary statistic everywhere-valid constraint. Note that this estimator is not admissible under the usual sense if p⩾5. Since the criterion of admissibility under the ancillary statistic everywhere-valid constraint is a reasonable one, that the constant coverage probability estimator has been commonly accepted is justified.     

The neyman accuracy and the wolfowitz accuracy of the stein type confidence interval for the disturbance variance

In this paper we consider the Neyman accuracy and the Wolfowitz accuracy of the Stein type improved confidence interval I?_S for the disturbance variance in a linear regression model. The Neyman accuracy is a measure related to the unbiasedness of a confidence interval, and the Wolfowitz accuracy is related to the closeness of the endpoints to the true parameter. We show that I?_S is not unbiased and give some numerical results for the Neyman accuracy. As for the Wolfowitz accuracy we derive the sufficient condition for I?_S to improve on the usual confidence interval under this criterion and show numerically that a large degree of improvement can be obtainted.     

Conditional interval estimation of the exponential location parameter following rejection of a pre-test

The conditional confidence interval for the location parameter of an exponential distribution following a preliminary test is investigated. The conditional confidence interval (CCI) may be shorter than the unconditional confidence interval (UCI) in contrast to the findings for the mean of a normal distribution by Meeks and D'Agostino (1983). The conditional coverage probability of the UCI is obtained by computing the coverage probability under the conditional probability density function. It is shown that the conditional coverage probability of the UCI is not uniformly greater than or less than the nominal level.     

Improved point and confidence interval estimators of mean response in simulation when control variates are used

In discrete event simulation, the method of control variates is often used to reduce the variance of estimation for the mean of the output response. In the present paper, it is shown that when three or more control variates are used, the usual linear regression estimator of the mean response is one of a large class of unbiased estimators, many of which have smaller variance than the usual estimator. In simulation studies using control variates, a confidence interval for the mean response is typically reported as well. Intervals with shorter width have been proposed using control variates in the literature. The present paper however develops confidence intervals which not only have shorter width but also have higher coverage probability than the usual confidence interval     

A generalized confidence limit for the reliability function of a two-parameter exponential distribution

Based on type II censored data, an exact lower confidence limit is constructed for the reliability function of a two-parameter exponential distribution, using the concept of a generalized confidence interval due to Weerahandi (J. Amer. Statist. Assoc. 88 (1993) 899). It is shown that the interval is exact, i.e., it provides the intended coverage. The confidence limit has to be numerically obtained; however, the required computations are simple and straightforward. An approximation is also developed for the confidence limit and its performance is numerically investigated. The numerical results show that compared to what is currently available, our approximation is more satisfactory in terms of providing the intended coverage, especially for small samples.     

Calibration Intervals in Linear Regression Models

Many of the existing methods of finding calibration intervals in simple linear regression rely on the inversion of prediction limits. In this article, we propose an alternative procedure which involves two stages. In the first stage, we find a confidence interval for the value of the explanatory variable which corresponds to the given future value of the response. In the second stage, we enlarge the confidence interval found in the first stage to form a confidence interval called, calibration interval, for the value of the explanatory variable which corresponds to the theoretical mean value of the future observation. In finding the confidence interval in the first stage, we have used the method based on hypothesis testing and percentile bootstrap. When the errors are normally distributed, the coverage probability of resulting calibration interval based on hypothesis testing is comparable to that of the classical calibration interval. In the case of non normal errors, the coverage probability of the calibration interval based on hypothesis testing is much closer to the target value than that of the calibration interval based on percentile bootstrap.     

Bayesian confidence interval for the risk ratio in a correlated 2 × 2 table with structural zero

This paper studies the construction of a Bayesian confidence interval for the risk ratio (RR) in a 2 × 2 table with structural zero. Under a Dirichlet prior distribution, the exact posterior distribution of the RR is derived, and tail-based interval is suggested for constructing Bayesian confidence interval. The frequentist performance of this confidence interval is investigated by simulation and compared with the score-based interval in terms of the mean coverage probability and mean expected width of the interval. An advantage of the Bayesian confidence interval is that it is well defined for all data structure and has shorter expected width. Our simulation shows that the Bayesian tail-based interval under Jeffreys’ prior performs as well as or better than the score-based confidence interval.     

Small confidence sets for the mean of a spherically symmetric distribution

Summary.  Suppose that X has a k -variate spherically symmetric distribution with mean vector θ and identity covariance matrix. We present two spherical confidence sets for θ , both centred at a positive part Stein estimator     . In the first, we obtain the radius by approximating the upper α -point of the sampling distribution of     by the first two non-zero terms of its Taylor series about the origin. We can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability, volume and conditional behaviour. In the second method, we find the radius by using a parametric bootstrap procedure. Here, even greater improvement in terms of volume over the usual confidence set is possible, at the expense of having a less explicit radius function. A real data example is provided, and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed.     

Confidence interval estimation for a linear contrast in intraclass correlation coefficients under unequal family sizes for several populations

Confidence intervals [based on F-distribution and (Z) standard normal distribution] for a linear contrast in intraclass correlation coefficients under unequal family sizes for several populations based on several independent multinormal samples have been proposed. It has been found that the confidence interval based on F-distribution consistently and reliably produced better results in terms of shorter average length of the interval than the confidence interval based on standard normal distribution for various combinations of intraclass correlation coefficient values. The coverage probability of the interval based on F-distribution is competitive with the coverage probability of the interval based on standard normal distribution. The interval based on F-distribution can be used for both small sample and large sample situations. An example with real life data has been presented.     

Comparison of the Stein and the usual estimators for the regression error variance under the Pitman nearness criterion when variables are omitted

This paper compares the Stein and the usual estimators of the error variance under the Pitman nearness (PN) criterion in a regression model which is mis-specified due to missing relevant explanatory variables. The exact expression of the PN-probability is derived and numerically evaluated. Contrary to the well-known result under mean squared errors (MSE), with the PN criterion the Stein variance estimator is uniformly dominated by the usual estimator when no relevant variables are excluded from the model. With an increased degree of model mis-specification, neither estimator strictly dominates the other. The authors are grateful to two anonymous referees for their valuable comments. Also, the first author is grateful to the Japan Society for the Promotion of Science for partial financial support.     

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.     

Generalized confidence interval for the slope in linear measurement error model

A generalized confidence interval for the slope parameter in linear measurement error model is proposed in this article, which is based on the relation between the slope of classical regression model and the measurement error model. The performance of the confidence interval estimation procedure is studied numerically through Monte Carlo simulation in terms of coverage probability and expected length.     

Confidence intervals centred on bootstrap smoothed estimators

Bootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data‐based model selection. A result of Efron in 2014 is a very convenient and widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. This approximation provides an easily computed guide to the accuracy of this estimator. In addition, Efron considered a confidence interval centred on the bootstrap smoothed estimator, with width proportional to the estimate of this approximation to the standard deviation. We evaluate this confidence interval in the scenario of two nested linear regression models, the full model and a simpler model, and a preliminary test of the null hypothesis that the simpler model is correct. We derive computationally convenient expressions for the ideal bootstrap smoothed estimator and the coverage probability and expected length of this confidence interval. In terms of coverage probability, this confidence interval outperforms the post‐model‐selection confidence interval with the same nominal coverage and based on the same preliminary test. We also compare the performance of the confidence interval centred on the bootstrap smoothed estimator, in terms of expected length, to the usual confidence interval, with the same minimum coverage probability, based on the full model.     

On improved interval estimation for the generalized variance

A confidence interval for the generalized variance of a matrix normal distribution with unknown mean is constructed which improves on the usual minimum size (i.e., minimum length or minimum ratio of endpoints) interval based on the sample generalized variance alone in terms of both coverage probability and size. The method is similar to the univariate case treated by Goutis and Casella (Ann. Statist. 19 (1991) 2015–2031).     

Bayesian Confidence Interval for the Ratio of Marginal Probabilities in the Matched-Pair Design

This article studies the construction of a Bayesian confidence interval for the ratio of marginal probabilities in matched-pair designs. Under a Dirichlet prior distribution, the exact posterior distribution of the ratio is derived. The tail confidence interval and the highest posterior density (HPD) interval are studied, and their frequentist performances are investigated by simulation in terms of mean coverage probability and mean expected length of the interval. An advantage of Bayesian confidence interval is that it is always well defined for any data structure and has shorter mean expected width. We also find that the Bayesian tail interval at Jeffreys prior performs as well as or better than the frequentist confidence intervals.     

A comment on sample size calculations for binomial confidence intervals

In this article we examine sample size calculations for a binomial proportion based on the confidence interval width of the Agresti–Coull, Wald and Wilson Score intervals. We pointed out that the commonly used methods based on known and fixed standard errors cannot guarantee the desired confidence interval width given a hypothesized proportion. Therefore, a new adjusted sample size calculation method was introduced, which is based on the conditional expectation of the width of the confidence interval given the hypothesized proportion. With the reduced sample size, the coverage probability can still maintain at the nominal level and is very competitive to the converge probability for the original sample size.     

On a theorem of Stein relating Bayesian and classical inferences in group models

Let a group G act on the sample space. This paper gives another proof of a theorem of Stein relating a group invariant family of posterior Bayesian probability regions to classical confidence regions when an appropriate prior is used. The example of the central multivariate normal distribution is discussed.