Confidence regions for parameters in linear models with additional information

Using given significant additional information it is possible to improve different confidence regions for the regression parameters in a linear model. Thereby, the given informations may concern the expectation and (or) the variance of the observations, and an improvement is possible in the sense of the decrease of the confidence regions' size. In particular it is possible to improve the so called confidence ellipsoids which are often used to estimate the considered parameters.     

An Optimal Confidence Region for the Largest and the Smallest Means from a Multivariate Normal Distribution

An optimal confidence region is proposed for obtaining the largest and the smallest means of a multivariate normal distribution with a common unknown variance and a non-negative correlation coefficient. The confidence region is compared to a class of asymmetric confidence regions, and the results show that it is uniformly better. Further, a design-oriented two-stage confidence region with a fixed width is also given. Finally, the optimal confidence region is applied to an experiment to measure the treatment effectiveness of a physical therapy with four independent groups and the result reveals that the proposed confidence region can provide some useful information.     

Interval Estimations for the Mean of the Generalized Poisson and Generalized Negative Binomial Distributions

In this article, we extend the Wald, score, skewness-corrected score, likelihood ratio, and mid-P intervals for the means of the generalized Poisson and generalized negative binomial distributions. These distributions are the members of the discrete version of the natural exponential family (NEF) with cubic variance function (CVF). Also, the coverage probabilities, the distal and mesial noncoverage probabilities, and the lengths of the proposed confidence intervals are estimated by means of a Monte Carlo simulation study. Finally, some practical examples are provided to show the applicability of the proposed intervals in applied studies.     

Asymptotic Confidence Limits for Performance Measures of a Repairable System with Imperfect Service Station

This article studies the asymptotic confidence limits for the steady-state availability, failure frequency, and mean time to failure of a repairable K-out-of-(M + S) system with M operating devices, S spares, and an imperfect service station that may be interrupted by a breakdown when it is repairing for the failed devices.     

Optimal Method for Realizing Two-Sided Inferences About a Linear Combination of Two Proportions

In order to reach the inference about a linear combination of two independent binomial proportions, various procedures exist (Wald's classic method, the exact, approximate, or maximized score methods, and the Newcombe-Zou method). This article defines and evaluates 25 different methods of inference, and selects the ones with the best behavior. In general terms, the optimal method is the classic Wald method applied to the data to which z ² _α/2/4 successes and z ² _α/2/4 failures are added (≈1 if α = 5%) if no sample proportion has a value of 0 or 1 (otherwise the added increase may be different).

Supplemental materials are available for this article. Go to the publisher's online edition of Communications in Statistics - Simulation and Computation to view the free supplemental file.     

Two-sample high-dimensional empirical likelihood

In this paper, we apply empirical likelihood for two-sample problems with growing high dimensionality. Our results are demonstrated for constructing confidence regions for the difference of the means of two p-dimensional samples and the difference in value between coefficients of two p-dimensional sample linear model. We show that empirical likelihood based estimator has the efficient property. That is, as p → ∞ for high-dimensional data, the limit distribution of the EL ratio statistic for the difference of the means of two samples and the difference in value between coefficients of two-sample linear model is asymptotic normal distribution. Furthermore, empirical likelihood (EL) gives efficient estimator for regression coefficients in linear models, and can be as efficient as a parametric approach. The performance of the proposed method is illustrated via numerical simulations.     

Some Small-Sample Properties of the Student t Confidence Intervals

For the two-sided Student t confidence interval for the mean of a normal distribution there is, for any sample size, a sufficiently large confidence level that ensures that the interval covers all the observations; there are also sufficiently small confidence levels guaranteeing, respectively, that (a) the interval does not cover all the observations and (b) the interval lies within the extreme observations. Necessary and sufficient conditions are also obtained for the width of the confidence interval to always exceed the sample range, as well as for the reverse inequality. Some implications of the results are discussed.     

Improved Confidence Statements for the Binomial Parameter

In the computation of two-sided confidence intervals for the binomial parameter p (using the binomial mass function), it is known that such intervals achieve a confidence coefficient that in general is not equal to the confidence level 1 – α, say. In this article we present some general results on the confidence coefficient and tabulate them for selected pairs (α, n = number of trials). We treat only the nominal equal tail probability case because it is the most commonly taught and used.     

Bootstrapping a Universal Pivot When Nuisance Parameters are Estimated

In complete samples from a continuous cumulative distribution with unknown parameters, it is known that various pivotal functions can be constructed by appealing to the probability integral transform. A pivotal function (or simply pivot) is a function of the data and parameters that has the property that its distribution is free of any unknown parameters. Pivotal functions play a key role in constructing confidence intervals and hypothesis tests. If there are nuisance parameters in addition to a parameter of interest, and consistent estimators of the nuisance parameters are available, then substituting them into the pivot can preserve the pivot property while altering the pivot distribution, or may instead create a function that is approximately a pivot in the sense that its asymptotic distribution is free of unknown parameters. In this latter case, bootstrapping has been shown to be an effective way of estimating its distribution accurately and constructing confidence intervals that have more accurate coverage probability in finite samples than those based on the asymptotic pivot distribution. In this article, one particular pivotal function based on the probability integral transform is considered when nuisance parameters are estimated, and the estimation of its distribution using parametric bootstrapping is examined. Applications to finding confidence intervals are emphasized. This material should be of interest to instructors of upper division and beginning graduate courses in mathematical statistics who wish to integrate bootstrapping into their lessons on interval estimation and the use of pivotal functions.

[Received November 2014. Revised August 2015.]     

Small Sample Confidence Intervals for the Odds Ratio

Abstract

Numerous methods—based on exact and asymptotic distributions—can be used to obtain confidence intervals for the odds ratio in 2 × 2 tables. We examine ten methods for generating these intervals based on coverage probability, closeness of coverage probability to target, and length of confidence intervals. Based on these criteria, Cornfield’s method, without the continuity correction, performed the best of the methods examined here. A drawback to use of this method is the significant possibility that the attained coverage probability will not meet the nominal confidence level. Use of a mid-P value greatly improves methods based on the “exact” distribution. When combined with the Wilson rule for selection of a rejection set, the resulting method is a procedure that performed very well. Crow’s method, with use of a mid-P, performed well, although it was only a slight improvement over the Wilson mid-P method. Its cumbersome calculations preclude its general acceptance. Woolf's (logit) method—with the Haldane–Anscombe correction— performed well, especially with regard to length of confidence intervals, and is recommended based on ease of computation.