Smallest confidence intervals for one binomial proportion

We specify three classes of one-sided and two-sided $1-\alpha$ confidence intervals with certain monotonicity and symmetry on the confidence limits for the probability of success, the parameter in a binomial distribution. For each class of one-sided confidence intervals the smallest interval, in the sense of the set inclusion, is obtained based on the direct analysis of coverage probability functions. A simple sufficient and necessary condition for the existence of the smallest two-sided confidence interval is provided and the smallest interval is derived if it exists. Thus the proposed intervals are uniformly most accurate, and have the uniformly minimum expected length as well.     

Improved simultaneous confidence intervals for linear contrasts and regression parameters

A well known method for obtaining conservative simultaneous confidence intervals for the K parameters in a linear regression model, or for K linear contrasts, is based on the percentage points of the Studentized maximum modulus distribution. From an inequality due to Sidak, conservative yet uniformly shorter confidence intervals would be possible if the percentage points of a particular form of the multivariate t distribution were available. The purpose of this paper is to provide the required percentage points. For K<8 the resulting confidence intervals can be substantially shorter.     

Optimal record-based statistical procedures for the two-parameter exponential distribution

There are a number of situations in which the experimental data observed are record statistics. In this paper, optimal confidence intervals as well as uniformly most powerful (MP) tests for one-sided alternatives are developed. Since a uniformly MP test for a two-sided alternative does not exist, generalized likelihood ratio and uniformly unbiased and invariant tests are derived for the two parameters of the exponential distribution based on record data. For illustrative purposes, a data set on the times between consecutive telephone calls to a company's switchboard is analysed using the proposed procedures. Finally, some open problems in this direction are pointed out.     

Comparisons of approximate confidence intervals for the smallest extreme value distribution simple linear regression model under time censoring

Exact confidence interval estimation for accelerated life regression models with censored smallest extreme value (or Weibull) data is often impractical. This paper evaluates the accuracy of approximate confidence intervals based on the asymptotic normality of the maximum likelihood estimator, the asymptotic X²distribution of the likelihood ratio statistic, mean and variance correction to the likelihood ratio statistic, and the so-called Bartlett correction to the likelihood ratio statistic. The Monte Carlo evaluations under various degrees of time censoring show that uncorrected likelihood ratio intervals are very accurate in situations with heavy censoring. The benefits of mean and variance correction to the likelihood ratio statistic are only realized with light or no censoring. Bartlett correction tends to result in conservative intervals. Intervals based on the asymptotic normality of maximum likelihood estimators are anticonservative and should be used with much caution.     

Exact uniformly most powerful postselection confidence distributions

A conditioning on the event of having selected one model from a set of possibly misspecified normal linear regression models leads to the construction of uniformly optimal conditional confidence distributions. They can be used for valid postselection inference. The constructed conditional confidence distributions are finite sample exact and encompass all information regarding the focus parameter in the selected model. This includes the construction of optimal postselection confidence intervals at all significance levels and uniformly most powerful hypothesis tests.     

Inferences on the lognormal mean for complete samples

In many engineering problems it is necessary to draw statistical inferences on the mean of a lognormal distribution based on a complete sample of observations. Statistical demonstration of mean time to repair (MTTR) is one example. Although optimum confidence intervals and hypothesis tests for the lognormal mean have been developed, they are difficult to use, requiring extensive tables and/or a computer. In this paper, simplified conservative methods for calculating confidence intervals or hypothesis tests for the lognormal mean are presented. In this paper, “conservative” refers to confidence intervals (hypothesis tests) whose infimum coverage probability (supremum probability of rejecting the null hypothesis taken over parameter values under the null hypothesis) equals the nominal level. The term “conservative” has obvious implications to confidence intervals (they are “wider” in some sense than their optimum or exact counterparts). Applying the term “conservative” to hypothesis tests should not be confusing if it is remembered that this implies that their equivalent confidence intervals are conservative. No implication of optimality is intended for these conservative procedures. It is emphasized that these are direct statistical inference methods for the lognormal mean, as opposed to the already well-known methods for the parameters of the underlying normal distribution. The method currently employed in MIL-STD-471A for statistical demonstration of MTTR is analyzed and compared to the new method in terms of asymptotic relative efficiency. The new methods are also compared to the optimum methods derived by Land (1971, 1973).     

Comparing two poisson parameters: what to do when the optimal isn't done

Suppose two Poisson processes with rates γ₁ and γ₂ are observed for fixed times t_l and t₂. This paper considers hypothesis tests and confidence intervals for the parameter ρ = γ₂/γ₁. Uniformly most powerful unbiased tests and uniformly most accurate unbiased confidence intervals exist for ρ, but they require randomization and so are not used in practice. Several alternative procedures have been proposed. In the context of one-sided hypothesis tests these procedures are reviewed and compared on numerical grounds and by use of the conditionality and repeated sampling principles. It is argued that a conditional binomial test which rejects with conditional level closest to but not necessarily less than, the nominal a is the most reasonable. This test is different from the usual conditional binomial test which rejects with conditional level closeset to but less than or equal to the nominal α Numerical results indicate that an approximate procedure based on the Poisson variance stabilizing transformation has properties similar to the preferred conditional binomial test. Values for λ₁ = t₁λ₁ required to achieve a specified power are considered. These results are also discussed in terms of test inversion to obtain confidence intervals.     

Efficient weighted bootstraps for the mean

The weighted bootstrap due to Mason and Newton (1992. Ann. Statist. 20, 1611–1624.) is applied to Studentized statistics in view of deriving efficient confidence intervals for the mean. First, we give conditions on the moments of the weights to ensure that the weighted bootstrap approximation leads to uniformly correct two-sided confidence intervals up to the rate O(n^−3/2). Then, we discuss the practical choice of the random weights in order to construct one-sided confidence intervals accurate up to O(n^−3/2) and two-sided confidence intervals up to higher orders. Simulations are given to illustrate the practical efficiency of our approach.     

Interval estimation of the mean in a two-stage nested model

The problem of constructing confidence intervals to estimate the mean in a two-stage nested model is considered. Several approximate intervals, which are based on both linear and nonlinear estimators of the mean are investigated. In particular, the method of bootstrap is used to correct the bias in the ‘usual’ variance of the nonlinear estimators. It is found that the intervals based on the nonlinear estimators did not achieve the nominal confidence coefficient for designs involving a small number of groups. Further, it turns out that the intervals are generally conservative, especially at small values of the intraclass correlation coefficient, and that the intervals based on the nonlinear estimators are more conservative than those based on the linear estimators. Compared with the others, the intervals based on the unweighted mean of the group means performed well in terms of coverage and length. For small values of the intraclass correlation coefficient, the ANOVA estimators of the variance components are recommended, otherwise the unweighted means estimator of the between groups variance component should be used. If one is fortunate enough to have control over the design, he is advised to increase the number of groups, as opposed to increasing group sizes, while avoiding groups of size one or two.     

Nonparametric confidence intervals for Tmax in sequence-stratified crossover studies

Tmax is the time associated with the maximum serum or plasma drug concentration achieved following a dose. While Tmax is continuous in theory, it is usually discrete in practice because it is equated to a nominal sampling time in the noncompartmental pharmacokinetics approach. For a 2-treatment crossover design, a Hodges-Lehmann method exists for a confidence interval on treatment differences. For appropriately designed crossover studies with more than two treatments, a new median-scaling method is proposed to obtain estimates and confidence intervals for treatment effects. A simulation study was done comparing this new method with two previously described rank-based nonparametric methods, a stratified ranks method and a signed ranks method due to Ohrvik. The Normal theory, a nonparametric confidence interval approach without adjustment for periods, and a nonparametric bootstrap method were also compared. Results show that less dense sampling and period effects cause increases in confidence interval length. The Normal theory method can be liberal (i.e. less than nominal coverage) if there is a true treatment effect. The nonparametric methods tend to be conservative with regard to coverage probability and among them the median-scaling method is least conservative and has shortest confidence intervals. The stratified ranks method was the most conservative and had very long confidence intervals. The bootstrap method was generally less conservative than the median-scaling method, but it tended to have longer confidence intervals. Overall, the median-scaling method had the best combination of coverage and confidence interval length. All methods performed adequately with respect to bias.     

On a confidence interval about a parameter estimated by a ratio of normal random variables

A confidence interval is geometrically constructed about a parameter estimated by the ratio of bivariate normal random variables. The resulting confidence interval is equivalent to that of Fieller's theorem. The geometric construction shown that such intervals are conservative. Bioassay examples are used to demonstrate the technique.     

A new confidence interval in errors-in-variables model with known error variance

This paper considers constructing a new confidence interval for the slope parameter in the structural errors-in-variables model with known error variance associated with the regressors. Existing confidence intervals are so severely affected by Gleser–Hwang effect that they are subject to have poor empirical coverage probabilities and unsatisfactory lengths. Moreover, these problems get worse with decreasing reliability ratio which also result in more frequent absence of some existing intervals. To ease these issues, this paper presents a fiducial generalized confidence interval which maintains the correct asymptotic coverage. Simulation results show that this fiducial interval is slightly conservative while often having average length comparable or shorter than the other methods. Finally, we illustrate these confidence intervals with two real data examples, and in the second example some existing intervals do not exist.     

Hypothesis Testing and Confidence Regions for the Mean Sojourn Time of an M/M/1 Queueing System

This article discusses testing hypotheses and confidence regions with correct levels for the mean sojourn time of an M/M/1 queueing system. The uniformly most powerful unbiased tests for three usual hypothesis testing problems are obtained and the corresponding p values are provided. Based on the duality between hypothesis tests and confidence sets, the uniformly most accurate confidence bounds are derived. A confidence interval with correct level is proposed.     

Conservative confidence intervals for the intraclass correlation coefficient for clustered binary data

Asymptotic approaches are traditionally used to calculate confidence intervals for intraclass correlation coefficient in a clustered binary study. When sample size is small to medium, or correlation or response rate is near the boundary, asymptotic intervals often do not have satisfactory performance with regard to coverage. We propose using the importance sampling method to construct the profile confidence limits for the intraclass correlation coefficient. Importance sampling is a simulation based approach to reduce the variance of the estimated parameter. Four existing asymptotic limits are used as statistical quantities for sample space ordering in the importance sampling method. Simulation studies are performed to evaluate the performance of the proposed accurate intervals with regard to coverage and interval width. Simulation results indicate that the accurate intervals based on the asymptotic limits by Fleiss and Cuzick generally have shorter width than others in many cases, while the accurate intervals based on Zou and Donner asymptotic limits outperform others when correlation and response rate are close to their boundaries.     

Using bootstrap method to evaluate the estimates of nicotine equivalents from linear mixed model and generalized estimating equation

Twenty-four-hour urinary excretion of nicotine equivalents, a biomarker for exposure to cigarette smoke, has been widely used in biomedical studies in recent years. Its accurate estimate is important for examining human exposure to tobacco smoke. The objective of this article is to compare the bootstrap confidence intervals of nicotine equivalents with the standard confidence intervals derived from linear mixed model (LMM) and generalized estimation equation. We use percentile bootstrap method because it has practical value for real-life application and it works well with nicotine data. To preserve the within-subject correlation of nicotine equivalents between repeated measures, we bootstrap the repeated measures of each subject as a vector. The results indicate that the bootstrapped estimates in most cases give better estimates than the LMM and generalized estimation equation without bootstrap.     

Confidence Intervals for Difference Between Two Poisson Rates

In this article, we develop four explicit asymptotic two-sided confidence intervals for the difference between two Poisson rates via a hybrid method. The basic idea of the proposed method is to estimate or recover the variances of the two Poisson rate estimates, which are required for constructing the confidence interval for the rate difference, from the confidence limits for the two individual Poisson rates. The basic building blocks of the approach are reliable confidence limits for the two individual Poisson rates. Four confidence interval estimators that have explicit solutions and good coverage levels are employed: the first normal with continuity correction, Rao score, Freeman and Tukey, and Jeffreys confidence intervals. Using simulation studies, we examine the performance of the four hybrid confidence intervals and compare them with three existing confidence intervals: the non-informative prior Bayes confidence interval, the t confidence interval based on Satterthwait's degrees of freedom, and the Bayes confidence interval based on Student's t confidence coefficient. Simulation results show that the proposed hybrid Freeman and Tukey, and the hybrid Jeffreys confidence intervals can be highly recommended because they outperform the others in terms of coverage probabilities and widths. The other methods tend to be too conservative and produce wider confidence intervals. The application of these confidence intervals are illustrated with three real data sets.     

A Simple Approximate Procedure for Constructing Binomial and Poisson Tolerance Intervals

The problems of constructing tolerance intervals for the binomial and Poisson distributions are considered. Closed-form approximate equal-tailed tolerance intervals (that control percentages in both tails) are proposed for both distributions. Exact coverage probabilities and expected widths are evaluated for the proposed equal-tailed tolerance intervals and the existing intervals. Furthermore, an adjustment to the nominal confidence level is suggested so that an equal-tailed tolerance interval can be used as a tolerance interval which includes a specified proportion of the population, but does not necessarily control percentages in both tails. Comparison of such coverage-adjusted tolerance intervals with respect to coverage probabilities and expected widths indicates that the closed-form approximate tolerance intervals are comparable with others, and less conservative, with minimum coverage probabilities close to the nominal level in most cases. The approximate tolerance intervals are simple and easy to compute using a calculator, and they can be recommended for practical applications. The methods are illustrated using two practical examples.     

Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

Multiple comparisons with a control for exponential location parameters under heteroscedasticity

In this paper, a new design-oriented two-stage two-sided simultaneous confidence intervals, for comparing several exponential populations with control population in terms of location parameters under heteroscedasticity, are proposed. If there is a prior information that the location parameter of k exponential populations are not less than the location parameter of control population, one-sided simultaneous confidence intervals provide more inferential sensitivity than two-sided simultaneous confidence intervals. But the two-sided simultaneous confidence intervals have advantages over the one-sided simultaneous confidence intervals as they provide both lower and upper bounds for the parameters of interest. The proposed design-oriented two-stage two-sided simultaneous confidence intervals provide the benefits of both the two-stage one-sided and two-sided simultaneous confidence intervals. When the additional sample at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, one-stage two-sided confidence intervals are proposed, which combine the advantages of one-stage one-sided and two-sided simultaneous confidence intervals. The critical constants are obtained using the techniques given in Lam [9,10]. These critical constant are compared with the critical constants obtained by Bonferroni inequality techniques and found that critical constant obtained by Lam [9,10] are less conservative than critical constants computed from the Bonferroni inequality technique. Implementation of the proposed simultaneous confidence intervals is demonstrated by a numerical example.     

Bayesian analysis for the two-parameter Pareto distribution based on record values and times

Doostparast and Balakrishnan (Pareto record-based analysis, Statistics, under review) recently developed optimal confidence intervals as well as uniformly most powerful tests for one- and two-sided hypotheses concerning shape and scale parameters, for the two-parameter Pareto distribution based on record data. In this paper, on the basis of record values and inter-record times from the two-parameter Pareto distribution, maximum-likelihood and Bayes estimators as well as credible regions are developed for the two parameters of the Pareto distribution. For illustrative purposes, a data set on annual wages of a sample of production-line workers in a large industrial firm is analysed using the proposed procedures.