Profile upper Confidence Limits from Discrete Data

When the data are discrete, standard approximate confidence limits often have coverage well below nominal for some parameter values. While ad hoc adjustments may largely solve this problem for particular cases, Kabaila & Lloyd (1997) gave a more systematic method of adjustment which leads to tight upper limits, which have coverage which is never below nominal and are as small as possible within a particular class. However, their computation for all but the simplest models is infeasible. This paper suggests modifying tight upper limits by an initial replacement of the unknown nuisance parameter vector by its profile maximum likelihood estimator. While the resulting limits no longer possess the optimal properties of tight limits exactly, the paper presents both numerical and theoretical evidence that the resulting coverage function is close to optimal. Moreover these profile upper limits are much (possibly many orders of magnitude) easier to compute than tight upper limits.     

TIGHT UPPER CONFIDENCE LIMITS FROM DISCRETE DATA

Consider the problem of finding an upper 1 –α confidence limit for a scalar parameter of interest ø in the presence of a nuisance parameter vector θ when the data are discrete. Approximate upper limits T may be found by approximating the relevant unknown finite sample distribution by its limiting distribution. Such approximate upper limits typically have coverage probabilities below, sometimes far below, 1 –α for certain values of (θ, ø). This paper remedies that defect by shifting the possible values t of T so that they are as small as possible subject both to the minimum coverage probability being greater than or equal to 1 –α, and to the shifted values being in the same order as the unshifted ts. The resulting upper limits are called ‘tight’. Under very weak and easily checked regularity conditions, a formula is developed for the tight upper limits.     

The M-Bayesian Credible Limits of the Reliability Derived from Binomial Distribution

This article introduces a new method, M-Bayesian credible limit method, to estimate reliability derived from binomial distribution, in the case of zero-failure data. Firstly, the definition of one-sided and two-sided M-Bayesian credible limits are provided, and moreover, formulas of one-sided and two-sided M-Bayesian credible limits are also provided. secondly, properties of one-sided and two-sided M-Bayesian credible limits are discussed, and we will see that the M-Bayesian credible limit method is superior to the corresponding classical confidence limit method. Finally, the new estimation method is applied to a numerical example. Through the example the efficiency and easiness of operation of this method are commended.     

Confidence intervals for the mean and a percentile based on zero-inflated lognormal data

The problems of estimating the mean and an upper percentile of a lognormal population with nonnegative values are considered. For estimating the mean of a such population based on data that include zeros, a simple confidence interval (CI) that is obtained by modifying Tian's [Inferences on the mean of zero-inflated lognormal data: the generalized variable approach. Stat Med. 2005;24:3223—3232] generalized CI, is proposed. A fiducial upper confidence limit (UCL) and a closed-form approximate UCL for an upper percentile are developed. Our simulation studies indicate that the proposed methods are very satisfactory in terms of coverage probability and precision, and better than existing methods for maintaining balanced tail error rates. The proposed CI and the UCL are simple and easy to calculate. All the methods considered are illustrated using samples of data involving airborne chlorine concentrations and data on diagnostic test costs.     

Buehler confidence limits and nesting

The Buehler 1 –α upper confidence limit is as small as possible, subject to the constraints that its coverage probability is at least 1 –α and that it is a non‐decreasing function of a pre‐specified statistic T. This confidence limit has important biostatistical and reliability applications. Previous research has examined the way the choice of T affects the efficiency of the Buehler 1 –α upper confidence limit for a given value of α. This paper considers how T should be chosen when the Buehler limit is to be computed for a range of values of α. If T is allowed to depend on α then the Buehler limit is not necessarily a non‐increasing function of α, i.e. the limit is ‘non‐nesting’. Furthermore, non‐nesting occurs in standard and practical examples. Therefore, if the limit is to be computed for a range [α_L, α_U]of values of α, this paper suggests that T should be a carefully chosen approximate 1 –α_L upper limit for θ. The choice leads to Buehler limits that have high statistical efficiency and are nesting.     

Computation of exact confidence intervals from discrete data using studentized test statistics

A new area of research interest is the computation of exact confidence limits or intervals for a scalar parameter of interest from discrete data by inverting a hypothesis test based on a studentized test statistic. See, for example, Chan and Zhang (1999), Agresti and Min (2001) and Agresti (2003) who deal with a difference of binomial probabilities and Agresti and Min (2002) who deal with an odds ratio. However, neither (1) a detailed analysis of the computational issues involved nor (2) a reliable method of computation that deals effectively with these issues is currently available. In this paper we solve these two problems for a very broad class of discrete data models. We suppose that the distribution of the data is determined by (,) where is a nuisance parameter vector. We also consider six different studentized test statistics. Our contributions to (1) are as follows. We show that the P-value resulting from the hypothesis test, considered as a function of the null-hypothesized value of , has both jump and drop discontinuities. Numerical examples are used to demonstrate that these discontinuities lead to the failure of simple-minded approaches to the computation of the confidence limit or interval. We also provide a new method for efficiently computing the set of all possible locations of these discontinuities. Our contribution to (2) is to provide a new and reliable method of computing the confidence limit or interval, based on the knowledge of this set.     

中国消费者信心影响因素实证分析

消费者信心是对消费者整体所表现出来的信心程度及其变动的一种测度。使用离散顺序选择模型对2009年第1季度中国大陆消费者信心指数的原始调查数据进行实证分析,结果显示：消费者对未来经济发展的信心受到其对未来就业、收入、生活和投资四个方面信心的显著影响。     

Confidence intervals for discrete distributions based on a single observation

Confidence intervals are developed for the mode of a discrete unimodal distribution in the case where only a single observation is available. These intervals are centered on either the observation, X, or a weighted average of X with a constant, b, chosen by the investigator. Intervals are derived for nonrestricted unimodal distributions, for unimodal distributions with a symmetry property, and for a family of two-sided truncated geometric distributions.     

Comparison of One-Sided Tolerance Limits in Unbalanced One-Way Random Models

Cox's discrete logistic model was extended to the study of the life table by Thompson (1977) to handle grouped survival data. Inferences about the effect of grouping are studies byMonte Carlo methods. The results show that the effect of grouping is not substantial. This approach is applied to the grouped data on liver cancer. The computer program developed for grouped censored data with continuous and indicator covariates is of practical importance and is available fromThe Ohio State University     

Exact Confidence Intervals,Based on the Z Statistic,for the Difference Between Two Proportions

We show that the confidence interval version of the extended exact unconditional Z test of Suissa and Shuster (1985) for testing the equality of two binomial proportions is due to general results of Buehler (1957), Sudakov and references cited there (1974), and Harris and Soms (1984). We apply these results to obtain exact unconditional confidence intervals for the difference between two proportions, deriving an explicit solution for the “best” outcome, make some comments on Buehler's (1957) method and give a numerical example. The Appendix contains a listing of the necessary FORTRAN programs.     

The Pearson Score Statistic for Multinomial-Poisson Models

The score statistic S² is commonly used for general likelihood-based inference. Pearson’s Chi-squared statistic X² = ∑(O ? E)²/E is ubiquitous in contingency table inference. Because tests and confidence intervals based on S² have been shown to work well in practice and theory and because X² has such a simple and intuitively appealing form, it is of interest to know when S² is identical to X² and when X² has an approximate Chi-squared distribution. Toward these ends, this paper gives a simple proof that S² = X² for the broad class of multinomial-Poisson distributions when the alternative hypothesis is unrestricted in a certain sense. This paper also gives a sufficient condition under which the null distribution of the Pearson score statistic is approximately Chi-squared. Several examples illustrate the utility of the results and counter-examples highlight the importance of the sufficient conditions of the results.     

Confidence bands for Brownian motion and applications to Monte Carlo simulation

Minimal area regions are constructed for Brownian paths and perturbed Brownian paths. While the theoretical optimal region cannot be obtained in closed form, we provide practical confidence regions based on numerical approximations and local time arguments. These regions are used to provide informal convergence assessments for both Monte Carlo and Markov Chain Monte Carlo experiments, via the Brownian asymptotic approximation of cumulative sums.     

Two-Sided M-Bayesian Credible Limits Method of Reliability Parameters and Its Applications

This article introduces a new method, named the two-sided M-Bayesian credible limits method, to estimate reliability parameters. It is especially suitable for situations of high reliability or zero-failure data. The definition, properties, and related formulas of the two-sided M-Bayesian credible limit are proposed. A real data set are also discussed. By means of an example we can see that the two-sided M-Bayesian credible limits method is efficient and easy to perform.     

Confidence interval estimation for number of patient‐years needed to treat

The number of patient‐years needed to treat (N_PYNT), also called the event‐based number needed to treat, to avoid one additional exacerbation has been reported in recently published respiratory trials, but the confidence intervals are not routinely reported. The challenge of constructing confidence intervals for N_PYNT is due to the fact that exacerbation data or count data in general are usually analyzed using Poisson‐based models such as Poisson or negative binomial regression and the rate ratio is the natural metric for between‐treatment comparison, while N_PYNT is based on rate difference, which is not usually calculated for those models. Therefore, the variance estimates from these analysis models are directly related to the rate ratio rather than the rate difference. In this paper, we propose several methods to construct confidence intervals for the N_PYNT, assuming that the event rates are estimated using Poisson or negative binomial regression models. The coverage property of the confidence intervals constructed with these methods is assessed by simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

Simultaneous Confidence Intervals for the Mean of Multivariate Poisson Distribution: A Comparison

Haibing (2009) proposed a procedure for successive comparisons between ordered treatment effects in one-way layout and showed that the proposed procedure has greater power than the procedure proposed by Lee and Spurrier (1995). Critical constants required for the proposed procedure were estimated using Monte Carlo simulation and few values of the constants were tabulated which limit the applications of the proposed procedure. In this article, a numerical method, using recursive integration methodology, is discussed to compute the critical constants which work efficiently for a large number of treatments and extensive values of critical constants are tabulated for the use of practitioners. Power comparisons of Haibing's and Lee and Spurrier's procedure is also discussed.     

A Note on Simultaneous Confidence Intervals for Ratio of Variances

An explicit formula for confidence intervals for ratios of variances of several populations is presented. The intervals are based on jackknife statistics and the critical point of the studentized range distribution. The asymptotic probability of coverage is not less than the nominal value provided that the distributions of the sampled populations belong to a location-scale family of probabilities with finite fourth moment.     

Comparison of Several Confidence Intervals for Median Residual Lifetime with Left-truncated and Right-censored Data

For left-truncated and right-censored data, the technique proposed by Brookmeyer and Crowley (1982) is extended to construct a point-wise confidence interval for median residual lifetime. This procedure is computationally simpler than the score type confidence interval in Jeong et al. (2008) and empirical likelihood ratio confidence interval in Zhou and Jeong (2011). Further, transformations of the estimator are applied to improve the approximation to the asymptotic distribution for small sample sizes. A simulation study is conducted to investigate the accuracy of these confidence intervals and the implementation of these confidence intervals to two real datasets is illustrated.     

Confidence intervals for assessing equivalence of two treatments with combined unilateral and bilateral data

Responses from the paired organs are generally highly correlated in bilateral studies, statistical procedures ignoring the correlation could lead to incorrect results. Note the intraclass correlation in the study of combined unilateral and bilateral outcomes; 11 confidence intervals (CIs) including 7 asymptotic CIs and 4 Bootstrap-resampling CIs for assessing the equivalence of 2 treatments are derived under Rosner''s correlated binary data model. Performance is evaluated with respect to the empirical coverage probability (ECP), the empirical coverage width (ECW) and the ratio of the mesial non-coverage probability to the non-coverage probability (RMNCP) via simulation studies. Simulation results show that (i) all CIs except for the Wald CI and the bias-corrected Bootstrap percentile CI generally produce satisfactory ECPs and hence are recommended; (ii) all CIs except for the bias-corrected Bootstrap percentile CI provide preferred RMNCPs and are more symmetrical; (iii) as the measurement of the dependence increases, the ECWs of all CIs except for the score CI and the profile likelihood CI show increasing patterns that look like linear, while there is no obvious pattern on the ECPs of all CIs except for the profile likelihood CI. A data set from an otolaryngologic study is used to illustrate the proposed methods.     

Confidence intervals for the difference of marginal probabilities in clustered matched-pair binary data

Although there are several available test statistics to assess the difference of marginal probabilities in clustered matched‐pair binary data, associated confidence intervals (CIs) are not readily available. Herein, the construction of corresponding CIs is proposed, and the performance of each CI is investigated. The results from Monte Carlo simulation study indicate that the proposed CIs perform well in maintaining the nominal coverage probability: for small to medium numbers of clusters, the intracluster correlation coefficient‐adjusted McNemar statistic and its associated Wald or Score CIs are preferred; however, this statistic becomes conservative when the number of clusters is larger so that alternative statistics and their associated CIs are preferred. In practice, a combination of the intracluster correlation coefficient‐adjusted McNemar statistic with an alternative statistic is recommended. To illustrate the practical application, a real clustered matched‐pair collection of data is used to illustrate testing the difference of marginal probabilities and constructing the associated CIs. Copyright © 2012 John Wiley & Sons, Ltd.