A procedure for approximate negative binomial tolerance intervals

In this article, we present a procedure for approximate negative binomial tolerance intervals. We utilize an approach that has been well-studied to approximate tolerance intervals for the binomial and Poisson settings, which is based on the confidence interval for the parameter in the respective distribution. A simulation study is performed to assess the coverage probabilities and expected widths of the tolerance intervals. The simulation study also compares eight different confidence interval approaches for the negative binomial proportions. We recommend using those in practice that perform the best based on our simulation results. The method is also illustrated using two real data examples.     

Confidence intervals for coefficients of variation in two-parameter exponential distributions

This article examines confidence intervals for the single coefficient of variation and the difference of coefficients of variation in the two-parameter exponential distributions, using the method of variance of estimates recovery (MOVER), the generalized confidence interval (GCI), and the asymptotic confidence interval (ACI). In simulation, the results indicate that coverage probabilities of the GCI maintain the nominal level in general. The MOVER performs well in terms of coverage probability when data only consist of positive values, but it has wider expected length. The coverage probabilities of the ACI satisfy the target for large sample sizes. We also illustrate our confidence intervals using a real-world example in the area of medical science.     

Equal-tailed and shortest Bayesian tolerance intervals based on exponential k-records

The problem of constructing the equal-tailed and shortest Bayesian tolerance intervals that control percentages in both tails of the exponential distribution based on k-record values is considered. Equal-tailed and shortest Bayesian tolerance factors are derived. Practical examples using real and simulated k-record values are given to illustrate the proposed results.     

Approximate tolerance intervals for the discrete Poisson–Lindley distribution

The Poisson–Lindley distribution is a compound discrete distribution that can be used as an alternative to other discrete distributions, like the negative binomial. This paper develops approximate one-sided and equal-tailed two-sided tolerance intervals for the Poisson–Lindley distribution. Practical applications of the Poisson–Lindley distribution frequently involve large samples, thus we utilize large-sample Wald confidence intervals in the construction of our tolerance intervals. A coverage study is presented to demonstrate the efficacy of the proposed tolerance intervals. The tolerance intervals are also demonstrated using two real data sets. The R code developed for our discussion is briefly highlighted and included in the tolerance package.     

Approximate and Fiducial Confidence Intervals for the Difference Between Two Binomial Proportions

The problem of estimating the difference between two binomial proportions is considered. Closed-form approximate confidence intervals (CIs) and a fiducial CI for the difference between proportions are proposed. The approximate CIs are simple to compute, and they perform better than the classical Wald CI in terms of coverage probabilities and precision. Numerical studies indicate that these approximate CIs can be used safely for practical applications under a simple condition. The fiducial CI is more accurate than the approximate CIs in terms of coverage probabilities. The fiducial CIs, the Newcombe CIs, and the Miettinen–Nurminen CIs are comparable in terms of coverage probabilities and precision. The interval estimation procedures are illustrated using two examples.     

Inference for functions of parameters in discrete distributions based on fiducial approach: Binomial and Poisson cases

In this article, we propose a simple method of constructing confidence intervals for a function of binomial success probabilities and for a function of Poisson means. The method involves finding an approximate fiducial quantity (FQ) for the parameters of interest. A FQ for a function of several parameters can be obtained by substitution. For the binomial case, the fiducial approach is illustrated for constructing confidence intervals for the relative risk and the ratio of odds. Fiducial inferential procedures are also provided for estimating functions of several Poisson parameters. In particular, fiducial inferential approach is illustrated for interval estimating the ratio of two Poisson means and for a weighted sum of several Poisson means. Simple approximations to the distributions of the FQs are also given for some problems. The merits of the procedures are evaluated by comparing them with those of existing asymptotic methods with respect to coverage probabilities, and in some cases, expected widths. Comparison studies indicate that the fiducial confidence intervals are very satisfactory, and they are comparable or better than some available asymptotic methods. The fiducial method is easy to use and is applicable to find confidence intervals for many commonly used summary indices. Some examples are used to illustrate and compare the results of fiducial approach with those of other available asymptotic methods.     

Improved closed-form prediction intervals for binomial and Poisson distributions

The problems of constructing prediction intervals for the binomial and Poisson distributions are considered. Available approximate, exact and conditional methods for both distributions are reviewed and compared. Simple approximate prediction intervals based on the joint distribution of the past samples and the future sample are proposed. Exact coverage studies and expected widths of prediction intervals show that the new prediction intervals are comparable to or better than the available ones in most cases. The methods are illustrated using two practical examples.     

Improved nonparametric tolerance intervals based on interpolated and extrapolated order statistics

The standard approach to construct nonparametric tolerance intervals is to use the appropriate order statistics, provided a minimum sample size requirement is met. However, it is well-known that this traditional approach is conservative with respect to the nominal level. One way to improve the coverage probabilities is to use interpolation. However, the extension to the case of two-sided tolerance intervals, as well as for the case when the minimum sample size requirement is not met, have not been studied. In this paper, an approach using linear interpolation is proposed for improving coverage probabilities for the two-sided setting. In the case when the minimum sample size requirement is not met, coverage probabilities are shown to improve by using linear extrapolation. A discussion about the effect on coverage probabilities and expected lengths when transforming the data is also presented. The applicability of this approach is demonstrated using three real data sets.     

Coverage Properties of Confidence Intervals for Generalized Additive Model Components

Abstract. We study the coverage properties of Bayesian confidence intervals for the smooth component functions of generalized additive models (GAMs) represented using any penalized regression spline approach. The intervals are the usual generalization of the intervals first proposed by Wahba and Silverman in 1983 and 1985, respectively, to the GAM component context. We present simulation evidence showing these intervals have close to nominal ‘across‐the‐function’ frequentist coverage probabilities, except when the truth is close to a straight line/plane function. We extend the argument introduced by Nychka in 1988 for univariate smoothing splines to explain these results. The theoretical argument suggests that close to nominal coverage probabilities can be achieved, provided that heavy oversmoothing is avoided, so that the bias is not too large a proportion of the sampling variability. The theoretical results allow us to derive alternative intervals from a purely frequentist point of view, and to explain the impact that the neglect of smoothing parameter variability has on confidence interval performance. They also suggest switching the target of inference for component‐wise intervals away from smooth components in the space of the GAM identifiability constraints.     

Tolerance intervals for symmetric location-scale families based on uncensored or censored samples

Exact methods for constructing two-sided tolerance intervals (TIs) and tolerance intervals that control percentages in both tails for a location-scale family of distributions are proposed. The proposed methods are illustrated by constructing TIs for a normal, logistic, and Laplace (double exponential) distributions based on type II singly censored samples. Factors for constructing one-sided and two-sided TIs for a logistic distribution are tabulated for the case of uncensored samples. Factors for constructing TIs based on censored samples for all three distributions are also tabulated. The factors for all cases are estimated by Monte Carlo simulation. An adjustment to the tolerance factors based on type II censored samples is proposed so that they can be used to find approximate TIs based on type I censored samples. Coverage studies of the approximate TIs based on type I censored samples indicate that the approximation is satisfactory as long as the proportion of censored observations is no more than 0.70. The methods are illustrated using some practical examples.     

Generalized confidence interval estimation for the mean of delta-lognormal distribution: an application to New Zealand trawl survey data

Highly skewed and non-negative data can often be modeled by the delta-lognormal distribution in fisheries research. However, the coverage probabilities of extant interval estimation procedures are less satisfactory in small sample sizes and highly skewed data. We propose a heuristic method of estimating confidence intervals for the mean of the delta-lognormal distribution. This heuristic method is an estimation based on asymptotic generalized pivotal quantity to construct generalized confidence interval for the mean of the delta-lognormal distribution. Simulation results show that the proposed interval estimation procedure yields satisfactory coverage probabilities, expected interval lengths and reasonable relative biases. Finally, the proposed method is employed in red cod densities data for a demonstration.     

Comparison of Confidence Intervals for a Poisson Mean – Further Considerations

I make recommendations in choosing a confidence interval for the Poisson mean, from twelve different methods, that are based on four general principles: actual coverage should closely match the nominal coverage; narrower expected widths of confidence intervals are better; the right and left non-coverage should be fairly balanced; and some investigators may prefer closed-form intervals. The interval chosen depends on the relative importance the investigator places on each of these principles. The confidence intervals are examined through graphs of their coverage probability, interval widths and shapes.     

An improved score-type confidence interval for stratified risk differences involving rare events

A stratified analysis of the differences in proportions has been widely employed in epidemiological research, social sciences, and drug development. It provides a useful framework for combining data across strata to produce a common effect. However, for rare events with incidence rates close to zero, popular confidence intervals for risk differences in a stratified analysis may not have appropriate coverage probabilities that approach the nominal confidence levels and the algorithms may fail to produce a valid confidence interval because of zero events in both the arms of a stratum. The main objective of this study is to evaluate the performance of certain methods commonly employed to construct confidence intervals for stratified risk differences when the response probabilities are close to a boundary value of zero or one. Additionally, we propose an improved stratified Miettinen–Nurminen confidence interval that exhibits a superior performance over standard methods while avoiding computational difficulties involving rare events. The proposed method can also be employed when the response probabilities are close to one.     

Bootstrap confidence intervals of CNpk for inverse Rayleigh and log-logistic distributions

In this article bootstrap confidence intervals of process capability index as suggested by Chen and Pearn [An application of non-normal process capability indices. Qual Reliab Eng Int. 1997;13:355–360] are studied through simulation when the underlying distributions are inverse Rayleigh and log-logistic distributions. The well-known maximum likelihood estimator is used to estimate the parameter. The bootstrap confidence intervals considered in this paper consists of various confidence intervals. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Application examples on two distributions for process capability indices are provided for practical use.     

Exact confidence limits for the probability of response in two-stage designs

In addition to point estimate for the probability of response in a two-stage design (e.g. Simon's two-stage design for binary endpoints), confidence limits should be computed and reported. The current method of inverting the p-value function to compute the confidence interval does not guarantee coverage probability in a two-stage setting. The existing exact approach to calculate one-sided limits is based on the overall number of responses to order the sample space. This approach could be conservative because many sample points have the same limits. We propose a new exact one-sided interval based on p-value for the sample space ordering. Exact intervals are computed by using binomial distributions directly, instead of a normal approximation. Both exact intervals preserve the nominal confidence level. The proposed exact interval based on the p-value generally performs better than the other exact interval with regard to expected length and simple average length of confidence intervals.     

A comparative study of confidence intervals for negative binomial proportion

In this paper, we investigate four existing and three new confidence interval estimators for the negative binomial proportion (i.e., proportion under inverse/negative binomial sampling). An extensive and systematic comparative study among these confidence interval estimators through Monte Carlo simulations is presented. The performance of these confidence intervals are evaluated in terms of their coverage probabilities and expected interval widths. Our simulation studies suggest that the confidence interval estimator based on saddlepoint approximation is more appealing for large coverage levels (e.g., nominal level≤1% ) whereas the score confidence interval estimator is more desirable for those commonly used coverage levels (e.g., nominal level>1% ). We illustrate these confidence interval construction methods with a real data set from a maternal congenital heart disease study.     

PROBLEMS WITH BINOMIAL TWO-SIDED TESTS AND THE ASSOCIATED CONFIDENCE INTERVALS

Confidence intervals for parameters of distributions with discrete sample spaces will be less conservative (i.e. have smaller coverage probabilities that are closer to the nominal level) when defined by inverting a test that does not require equal probability in each tail. However, the P‐value obtained from such tests can exhibit undesirable properties, which in turn result in undesirable properties in the associated confidence intervals. We illustrate these difficulties using P‐values for binomial proportions and the difference between binomial proportions.     

Shortest tolerance intervals controlling both tails of the exponential distribution based on record values

This paper deals with computing shortest width tolerance intervals controlling both tails of the exponential distribution on the basis of record values. Equal-tailed and shortest tolerance factors are derived. The expected widths of these tolerance intervals are evaluated via a Monte Carlo simulation study. Finally, two illustrative examples are also included.     

Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions

One of the indicators for evaluating the capability of a process is the process capability index. In this article, bootstrap confidence intervals of the generalized process capability index (GPCI) proposed by Maiti et al. are studied through simulation, when the underlying distributions are Lindley and Power Lindley distributions. The maximum likelihood method is used to estimate the parameters of the models. Three bootstrap confidence intervals namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile bootstrap (BCPB) are considered for obtaining confidence intervals of GPCI. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average width of the bootstrap confidence intervals. Simulation results show that the estimated coverage probabilities of the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval get closer to the nominal confidence level than those of the standard bootstrap confidence interval. Finally, three real datasets are analyzed for illustrative purposes.     

Mean-Minimum Exact Confidence Intervals

This article introduces mean-minimum (MM) exact confidence intervals for a binomial probability. These intervals guarantee that both the mean and the minimum frequentist coverage never drop below specified values. For example, an MM 95[93]% interval has mean coverage at least 95% and minimum coverage at least 93%. In the conventional sense, such an interval can be viewed as an exact 93% interval that has mean coverage at least 95% or it can be viewed as an approximate 95% interval that has minimum coverage at least 93%. Graphical and numerical summaries of coverage and expected length suggest that the Blaker-based MM exact interval is an attractive alternative to, even an improvement over, commonly recommended approximate and exact intervals, including the Agresti–Coull approximate interval, the Clopper–Pearson (CP) exact interval, and the more recently recommended CP-, Blaker-, and Sterne-based mean-coverage-adjusted approximate intervals.