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.     

Mid-P confidence intervals based on the likelihood ratio for proportions estimated by group testing

Group testing has been used in many fields of study to estimate proportions. When groups are of different size, the derivation of exact confidence intervals is complicated by the lack of a unique ordering of the event space. An exact interval estimation method is described here, in which outcomes are ordered according to a likelihood ratio statistic. The method is compared with another exact method, in which outcomes are ordered by their associated MLE. Plots of the P‐value against the proportion are useful in examining the properties of the methods. Coverage provided by the intervals is assessed using several realistic grouptesting procedures. The method based on the likelihood ratio, with a mid‐P correction, is shown to give very good coverage in terms of closeness to the nominal level, and is recommended for this type of problem.     

Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation

In this article, we point out some interesting relations between the exact test and the score test for a binomial proportion p. Based on the properties of the tests, we propose some approximate as well as exact methods of computing sample sizes required for the tests to attain a specified power. Sample sizes required for the tests are tabulated for various values of p to attain a power of 0.80 at level 0.05. We also propose approximate and exact methods of computing sample sizes needed to construct confidence intervals with a given precision. Using the proposed exact methods, sample sizes required to construct 95% confidence intervals with various precisions are tabulated for p = .05(.05).5. The approximate methods for computing sample sizes for score confidence intervals are very satisfactory and the results coincide with those of the exact methods for many cases.     

Coverage‐adjusted Confidence Intervals for a Binomial Proportion

We consider the classic problem of interval estimation of a proportion p based on binomial sampling. The ‘exact’ Clopper–Pearson confidence interval for p is known to be unnecessarily conservative. We propose coverage adjustments of the Clopper–Pearson interval that incorporate prior or posterior beliefs into the interval. Using heatmap‐type plots for comparing confidence intervals, we show that the coverage‐adjusted intervals have satisfying coverage and shorter expected lengths than competing intervals found in the literature.     

Exact Likelihood Inference for k Exponential Populations Under Joint Type-II Censoring

In this paper, when a jointly Type-II censored sample arising from k independent exponential populations is available, the conditional MLEs of the k exponential mean parameters are derived. The moment generating functions and the exact densities of these MLEs are obtained using which exact confidence intervals are developed for the parameters. Moreover, approximate confidence intervals based on the asymptotic normality of the MLEs and credible confidence regions from a Bayesian viewpoint are also discussed. An empirical comparison of the exact, approximate, bootstrap, and Bayesian intervals is also made in terms of coverage probabilities. Finally, an example is presented in order to illustrate all the methods of inference developed here.     

Comparison of Poisson Confidence Intervals

Abstract

The standard method of obtaining a two-sided confidence interval for the Poisson mean produces an interval which is exact but can be shortened without violating the minimum coverage requirement. We classify the intervals proposed as alternatives to the standard method interval. We carry out the classification using two desirable properties of two-sided confidence intervals.     

Usual and Shortest Confidence Intervals on Odds Ratios from Logistic Regression

Based on the large-sample normal distribution of the sample log odds ratio and its asymptotic variance from maximum likelihood logistic regression, shortest 95% confidence intervals for the odds ratio are developed. Although the usual confidence interval on the odds ratio is unbiased, the shortest interval is not. That is, while covering the true odds ratio with the stated probability, the shortest interval covers some values below the true odds ratio with higher probability. The upper and lower limits of the shortest interval are shifted to the left of those of the usual interval, with greater shifts in the upper limits. With the log odds model γ + Xβ, in which X is binary, simulation studies showed that the approximate average percent difference in length is 7.4% for n (sample size) = 100, and 3.8% for n = 200. Precise estimates of the covering probabilities of the two types of intervals were obtained from simulation studies, and are compared graphically. For odds ratio estimates greater (less) than one, shortest intervals are more (less) likely to include one than are the usual intervals. The usual intervals are likelihood-based and the shortest intervals are not. The usual intervals have minimum expected length among the class of unbiased intervals. Shortest intervals do not provide important advantages over the usual intervals, which we recommend for practical use.     

Confidence intervals for the difference in paired Youden indices

Comparison of accuracy between two diagnostic tests can be implemented by investigating the difference in paired Youden indices. However, few literature articles have discussed the inferences for the difference in paired Youden indices. In this paper, we propose an exact confidence interval for the difference in paired Youden indices based on the generalized pivotal quantities. For comparison, the maximum likelihood estimate‐based interval and a bootstrap‐based interval are also included in the study for the difference in paired Youden indices. Abundant simulation studies are conducted to compare the relative performance of these intervals by evaluating the coverage probability and average interval length. Our simulation results demonstrate that the exact confidence interval outperforms the other two intervals even with small sample size when the underlying distributions are normal. A real application is also used to illustrate the proposed intervals. Copyright © 2012 John Wiley & Sons, Ltd.     

Interval estimation for a binomial proportion: a bootstrap approach

This paper discusses the classic but still current problem of interval estimation of a binomial proportion. Bootstrap methods are presented for constructing such confidence intervals in a routine, automatic way. Three confidence intervals for a binomial proportion are compared and studied by means of a simulation study, namely: the Wald confidence interval, the Agresti–Coull interval and the bootstrap-t interval. A new confidence interval, the Agresti–Coull interval with bootstrap critical values, is also introduced and its good behaviour related to the average coverage probability is established by means of simulations.     

A Comparison of Large-Sample Confidence Interval Methods for the Difference of Two Binomial Probabilities

A simulation study was done to compare seven confidence interval methods, based on the normal approximation, for the difference of two binomial probabilities. Cases considered included minimum expected cell sizes ranging from 2 to 15 and smallest group sizes (NMIN) ranging from 6 to 100. Our recommendation is to use a continuity correction of 1/(2 NMIN) combined with the use of (N ? 1) rather than N in the estimate of the standard error. For all of the cases considered with minimum expected cell size of at least 3, this method gave coverage probabilities close to or greater than the nominal 90% and 95%. The Yates method is also acceptable, but it is slightly more conservative. At the other extreme, the usual method (with no continuity correction) does not provide adequate coverage even at the larger sample sizes. For the 99% intervals, our recommended method and the Yates correction performed equally well and are reasonable for minimum expected cell sizes of at least 5. None of the methods performed consistently well for a minimum expected cell size of 2.     

Exact average coverage probabilities and confidence coefficients of confidence intervals for discrete distributions

For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf  _θ P _θ (θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions are proposed. With these methodologies, both exact values can be derived.     

On Ranges of Confidence Coefficients for Confidence Intervals on Variance Components

The among variance component in the balanced one-factor nested components-of-variance model is of interest in many fields of application. Except for an artificial method that uses a set of random numbers which is of no use in practical situations, an exact-size confidence interval on the among variance has not yet been derived. This paper provides a detailed comparison of three approximate confidence intervals which possess certain desired properties and have been shown to be the better methods among many available approximate procedures. Specifically, the minimum and the maximum of the confidence coefficients for the one- and two-sided intervals of each method are obtained. The expected lengths of the intervals are also compared.     

Conservative confidence intervals for a single parameter

The method of constructing confidence intervals from hypothesis tests is studied in the case in which there is a single unknown parameter and is proved to provide confidence intervals with coverage probability that is at least the nominal level. The confidence intervals obtained by the method in several different contexts are seen to compare favorably with confidence intervals obtained by traditional methods. The traditional intervals are seen to have coverage probability less than the nominal level in several instances, This method can be applied to all confidence interval problems and reduces to the traditional method when an exact pivotal statistic is known.     

Exact Likelihood Inference for k Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of k competing products with regard to their reliability. In this paper, when a joint progressively Type-II censored sample arising from k independent exponential populations is available, the conditional MLEs of the k exponential mean parameters are derived. Their conditional moment generating functions and exact densities are obtained, using which exact confidence intervals are developed for the parameters. Moreover, approximate confidence intervals based on the asymptotic normality of the MLEs and credible confidence regions from a Bayesian viewpoint are discussed. An empirical evaluation of the exact, approximate, bootstrap, and Bayesian intervals is also made in terms of coverage probabilities and average widths. Finally, an example is presented in order to illustrate all the methods of inference developed here.     

Exact confidence intervals for randomized response strategies

For surveys with sensitive questions, randomized response sampling strategies are often used to increase the response rate and encourage participants to provide the truth of the question while participants' privacy and confidentiality are protected. The proportion of responding ‘yes’ to the sensitive question is the parameter of interest. Asymptotic confidence intervals for this proportion are calculated from the limiting distribution of the test statistic, and are traditionally used in practice for statistical inference. It is well known that these intervals do not guarantee the coverage probability. For this reason, we apply the exact approach, adjusting the critical value as in [10 J. Frey and A. Pérez, Exact binomial confidence intervals for randomized response, Amer. Statist.66 (2012), pp. 8–15. Available at http://dx.doi.org/10.1080/00031305.2012.663680.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], to construct the exact confidence interval of the proportion based on the likelihood ratio test and three Wilson-type tests. Two randomized response sampling strategies are studied: the Warner model and the unrelated model. The exact interval based on the likelihood ratio test has shorter average length than others when the probability of the sensitive question is low. Exact Wilson intervals have good performance in other cases. A real example from a survey study is utilized to illustrate the application of these exact intervals.     

Estimation Bias in the First-Order Autoregressive Model and Its Impact on Predictions and Prediction Intervals

The least squares estimate of the autoregressive coefficient in the AR(1) model is known to be biased towards zero, especially for parameters close to the stationarity boundary. Several methods for correcting the autoregressive parameter estimate for the bias have been suggested. Using simulations, we study the bias and the mean square error of the least squares estimate and the bias-corrections proposed by Kendall and Quenouille.

We also study the mean square forecast error and the coverage of the 95% prediction interval when using the biased least squares estimate or one of its bias-corrected versions. We find that the estimation bias matters little for point forecasts, but that it affects the coverage of the prediction intervals. Prediction intervals for forecasts more than one step ahead, when calculated with the biased least squares estimate, are too narrow.     

The Fate Worse than Death and other Curiosities and Stupidities

The problem of finding confidence intervals for the success parameter of a binomial experiment has a long history, and a myriad of procedures have been developed. Most exploit the duality between hypothesis testing and confidence regions and are typically based on large sample approximations. We instead employ a direct approach that attempts to determine the optimal coverage probability function a binomial confidence procedure can have from the exact underlying binomial distributions, which in turn defines the associated procedure. We show that a graphical perspective provides much insight into the problem. Both procedures whose coverage never falls below the declared confidence level and those that achieve that level only approximately are analyzed. We introduce the Length/Coverage Optimal method, a variant of Sterne's procedure that minimizes average length while maximizing coverage among all length minimizing procedures, and show that it is superior in important ways to existing procedures.     

Exact Non-Parametric Confidence, Prediction and Tolerance Intervals with Progressive Type-II Censoring

Abstract.  This article extends recent results [Scand. J. Statist. 28 (2001) 699] about exact non-parametric inferences based on order statistics with progressive type-II censoring. The extension lies in that non-parametric inferences are now covered where the dependence between involved order statistics cannot be circumvented. These inferences include: (a) tolerance intervals containing at least a specified proportion of the parent distribution, (b) prediction intervals containing at least a specified number of observations in a future sample, and (c) outer and/or inner confidence intervals for a quantile interval of the parent distribution. The inferences are valid for any parent distribution with continuous distribution function. The key result shows how the probability of an event involving k dependent order statistics that are observable/uncensored with progressive type-II censoring can be represented as a mixture with known weights of corresponding probabilities involving k dependent ordinary order statistics. Further applications/developments concerning exact Kolmogorov-type confidence regions are indicated.     

βExpectation and β-content tolerance intervals for dependent observations

Methods of constructing exact tolerance intervals (β-expectation and β-content) for independent observations are well known. For the case of dependent observations, obtaining exact results is not possible. In this article we provide an approximate method of constructing β-expectation tolerance intervals via a Taylor series expansion. Examples of independent observations are considered to compare the intervals constructed with those obtained by the exact method. For the case of non-stationary type processes we have proposed a method of constructing approximate β-content tolerance intervals. Once again an example is given to illustrate the results.