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We develop two new approximate confidence interval methods for estimating a population proportion using balanced ranked-set sampling (RSS). Unlike existing RSS-based methods, the new methods control the coverage probability well not just under perfect rankings, but also under imperfect rankings. One method uses a Wilson-type interval, and the other is based on making a mid-P adjustment to a Clopper–Pearson-type interval. Both methods rely on a new maximum-likelihood-based method for estimating the proportions in the judgment strata when the overall proportion is given, and both can be computed even for large sample sizes.

In this paper, a robust interval estimator for the classical process capability index (Cp) based on the modified trimmed standard deviation (MTSD = ST ∗ ) is considered under both normal and non-normal distributions. The performance of the newly proposed process capability index interval estimator over the existing method is compared using a simulation study. As a performance criterion, we consider both simulated coverage probability and average width. Simulation results evident that the proposed confidence interval based on the robust estimator performed well for most of cases. For illustration purposes, two real-life data from industry are analyzed which supported our simulation results to some extent. As a result, the proposed method can be recommend to be used by the practitioners in various fields of industry, engineering and physical sciences.

In this paper, three robust confidence intervals are proposed as alternatives to the Student t confidence interval. The performance of these intervals was compared through a simulation study shows that Qn-t confidence interval performs the best and it is as good as Student’s t confidence interval. Real-life data was used for illustration and performing a comparison that support the findings obtained from the simulation study.

Statistical Model Checking (SMC) is an approximate verification method that overcomes the state space explosion problem for probabilistic systems by Monte Carlo simulations. Simulations might, however, be costly if many samples are required. It is thus necessary to implement efficient algorithms to reduce the sample size while preserving precision and accuracy. In the literature, some sequential schemes have been provided for the estimation of property occurrence based on predefined confidence and absolute or relative error. Nevertheless, these algorithms remain conservative and may result in huge sample sizes if the required precision standards are demanding. In this article, we compare some useful bounds and some sequential methods. We propose outperforming and rigorous alternative schemes based on Massart bounds and robust confidence intervals. Our theoretical and empirical analyses show that our proposal reduces the sample size while providing the required guarantees on error bounds.

For a symmetric unimodal random variable with known mode and variance, we construct a tight upper bound for the probability of falling outside an arbitrarily specified interval. We define the distribution on which the bound is achieved. In the special case when the distance between mode and the middle of the interval is less than 29% of its radius, the resulting bound is determined by the Gauss inequality. We have applied our result to the construction of robust confidence intervals.

Traditional confidence intervals are useful in engineering because they offer a guarantee of statistical performance through repeated use. However, it is difficult to employ them consistently in analyses and assessments because it is not clear how to propagate them through mathematical calculations. Confidence structures (c-boxes) generalize confidence distributions and provide an interpretation by which confidence intervals at any confidence level can be specified for a parameter of interest. C-boxes can be used in calculations using the standard methods of probability bounds analysis and yield results that also admit the confidence interpretation. Thus analysts using them can now literally compute with confidence. We illustrate the calculation and use of c-boxes for some elementary inference problems and describe R functions to compute them and some Monte Carlo simulations demonstrating the coverage performance of the c-boxes and calculations based on them.

Maximum likelihood (ML) estimators of the model parameters in multiple linear regression are obtained using genetic algorithm (GA) when the distribution of the error terms is long-tailed symmetric. We compare the efficiencies of the ML estimators obtained using GA with the corresponding ML estimators obtained using other iterative techniques via an extensive Monte Carlo simulation study. Robust confidence intervals based on modified ML estimators are used as the search space in GA. Our simulation study shows that GA outperforms traditional algorithms in most cases. Therefore, we suggest using GA to obtain the ML estimates of the multiple linear regression model parameters when the distribution of the error terms is LTS. Finally, real data of the Covid-19 pandemic, a global health crisis in early 2020, is presented for illustrative purposes.

In this paper, a robust analysis of nitrogen dioxide (NO2) concentration measurements taken at Belisario station (Quito, Ecuador) was performed. The data used for the analysis constitute a set of measurements taken from 1 January 2008 to 31 December 2019. Furthermore, the analysis was carried out in a robust way, defining variables that represent years, months, days and hours, and classifying these variables based on estimates of the central tendency and dispersion of the data. The estimators used here were classic, nonparametric, based on a bootstrap method, and robust. Additionally, confidence intervals based on these estimators were built, and these intervals were used to categorize the variables under study. The results of this research showed that the NO2 concentration at Belisario station is not harmful to humans. Moreover, it was shown that this concentration tends to be stable across the years, changes slightly during the days of the week, and varies greatly when analyzed by months and hours of the day. Here, the precision provided by both nonparametric and robust statistical methods served to comprehensively proof the aforementioned. Finally, it can be concluded that the city of Quito is progressing on the right path in terms of improving air quality, because it has been shown that there is a decreasing tendency in the NO2 concentration across the years. In addition, according to the Quito Air Quality Index, most of the observations are in either the desirable level or acceptable level of air pollution, and the number of observations that are in the desirable level of air pollution increases across the years.

There is often a need to estimate the characteristics of epidemics or seasonality from infectious disease data. For instance, accurately estimating the start and end date of respiratory syncytial virus (RSV) epidemics can be used to optimize the initiation of prophylactic medication. Many widely-used methods for describing these characteristics begin with a regression model fit to a time series of disease incidence. The fitted model is then often used to calculate the quantities of interest. Calculation of these quantities from the fitted regression model typically involves combining together different components of the fitted model, and consequently only point estimates (rather than measures of uncertainty) of those quantities can be made in a straightforward way. Motivated by attempts to estimate the optimal timing of prophylaxis for RSV, we developed a general method for obtaining confidence intervals for characteristics of seasonal and sporadic infectious disease outbreaks. To do this, we use multivariate sampling of a generalized additive model with penalized basis splines. Our approach provides robust confidence intervals regardless of the complexity of the calculations of the outcome measures, and it generalizes to other systems (including outbreaks of other infectious diseases). Here we present our general approach, its application to RSV, and an R package that provides a convenient interface for conducting and validating this type of analysis in other areas.

This paper provides guidelines for determining average grain sizes with an optimal level of statistical adequacy and determine the best methods for estimating confidence intervals in grain size studies. I used Monte Carlo simulations and relative efficiency curves to test which central tendency measure (average) and which method of confidence interval estimation perform best depending on the lognormal shape and sample size. Results from synthetic populations were then compared and validated using actual grain size populations and random sampling methods. Overall, the geometric mean and the confidence intervals for the geometric mean provide the best balance between efficiency and robustness. The median is the best alternative when data contamination is an issue. Confidence intervals for the arithmetic mean in actual grain size populations are problematic regardless of the method used, especially for small sample sizes (n

Current approaches to antibiotic dose determination in critically ill patients requiring renal replacement therapy are primarily based on the assessment of highly heterogeneous data from small number of patients. The standard modelling approaches limit the scope of constructing robust confidence boundaries of the distribution of pharmacokinetics (PK) parameters, especially when the evaluation of possible association of demographic and clinical factors at different levels of the distribution of drug clearance is of interest. Commonly used compartmental models generally construct the inferences through a linear or non-linear mean regression, which is inadequate when the distribution is skewed, multi-modal or effected by atypical observation. In this study, we discuss the statistical challenges in robust estimation of the confidence boundaries of the PK parameters in the presence of highly heterogenous patient characteristics. A novel stepwise approach to evaluate the confidence boundaries of PK parameters is proposed by combining PK modelling with mixed-effects quantile regression (MEQR) methods. This method allows the assessment demographic and clinical factors’ effects at any arbitrary quantiles of the outcome of interest, without restricting assumptions on the distributions. The MEQR approach allows us to investigate if the levels of association of the covariates are different at low, medium or high concentration. This methodological assessment is deemed as a background initial approach to support the development of a class of statistical algorithm in constructing robust confidence intervals of PK parameters which can be used for developing an optimised antibiotic dosing guideline for critically ill patients requiring renal replacement therapy.

Statistical inference about parameters should depend on raw data only through sufficient statistics—the well known sufficiency principle. In particular, inference should depend on minimal sufficien...

In this paper, a regression semi-parametric model is considered where responses are assumed to be missing at random. From the empirical likelihood function defined based on the rank-based estimating equation, robust confidence intervals/regions of the true regression coefficient are derived. Monte Carlo simulation experiments show that the proposed approach provides more accurate confidence intervals/regions compared to its normal approximation counterpart under different model error structure. The approach is also compared with the least squares approach, and its superiority is shown whenever the error distribution in the simulation study is heavy tailed or contaminated. Finally, a real data example is given to illustrate our proposed method.

Under a reasonable restriction, we create the minimum volume confidence set for location and scale parameters of the exponential distribution. Compared to existing methods, none of which has a mini...

We propose a method that integrates bootstrap into the forward search algorithm in the construction of robust confidence intervals for elements of the eigenvectors of the correlation matrix in the ...

Ordinal coefficient alpha is a newly derived non-parametric reliability estimate. As with any point estimate, ordinal coefficient alpha is merely an estimate of a population parameter and tends to vary from sample to sample. Researchers report the confidence interval to provide readers with the amount of precision obtained. Several methods with differing computational approaches exist for confidence interval estimation for alpha, including the Fisher, Feldt, Bonner, and Hakstian and Whalen (HW) techniques. Overall, coverage rates for the various methods were unacceptably low with the Fisher method as the highest performer at 62%. Because of the poor performance across all four confidence interval methods, a need exists to develop a method which works well for ordinal coefficient alpha.

An empirical Bayes confidence interval has high user demand in many applications. In particular, the second-order empirical Bayes confidence interval, the coverage error of which is of the third order for large number of areas, is widely used in small area estimation when the sample size within each area is not large enough to make reliable direct estimates based on a design-based approach. Yoshimori and Lahiri (2014a) proposed a new type of confidence interval, called the second-order efficient empirical Bayes confidence interval, whose length is less than that of the direct confidence interval based on the design-based approach. However, this interval still has some disadvantages: (i) it is hard to use when at least one leverage value is high; (ii) many iterations tend to be required to obtain the estimators of one global model variance parameter as the number of areas getting larger, due to the area-specific adjustment factor. To prevent such issues, this paper proposes, as never done before, a more efficient confidence interval to allow for high leverage and reduce the number of iterations for large number of areas, by adopting a non-area-specific adjustment factor, maintaining the existing desired properties. We also reveal the relationship between the general adjustment factor and the measure of uncertainty of the empirical Bayes estimator to create a second-order confidence interval. Moreover, we present two simulation studies and a real data analysis to show the efficiency of this confidence interval., Comment: 18 pages, 2 tables

A Bayesian network represents a causal relationship among random variables using conditional probabilities. Because of limited resources and sampling uncertainty, the estimated probabilities have b...

Inference after two-stage single-arm designs with binary endpoint is challenging due to the nonunique ordering of the sampling space in multistage designs. We illustrate the problem of specifying test-compatible confidence intervals for designs with nonconstant second-stage sample size and present two approaches that guarantee confidence intervals consistent with the test decision. Firstly, we extend the well-known Clopper-Pearson approach of inverting a family of two-sided hypothesis tests from the group-sequential case to designs with fully adaptive sample size. Test compatibility is achieved by using a sample space ordering that is derived from a test-compatible estimator. The resulting confidence intervals tend to be conservative but assure the nominal coverage probability. In order to assess the possibility of further improving these confidence intervals, we pursue a direct optimization approach minimizing the mean width of the confidence intervals. While the latter approach produces more stable coverage probabilities, it is also slightly anti-conservative and yields only negligible improvements in mean width. We conclude that the Clopper-Pearson-type confidence intervals based on a test-compatible estimator are the best choice if the nominal coverage probability is not to be undershot and compatibility of test decision and confidence interval is to be preserved.

When the multivariate normality assumption is violated in structural equation modeling, a leading remedy involves estimation via normal theory maximum likelihood with robust corrections to standard...

The interval between two prespecified order statistics of a sample provides a distribution-free confidence interval for a population quantile. However, due to discreteness, only a small set of exact coverage probabilities is available. Interpolated confidence intervals are designed to expand the set of available coverage probabilities. However, we show here that the infimum of the coverage probability for an interpolated confidence interval is either the coverage probability for the inner interval or the coverage probability obtained by removing the more likely of the two extreme subintervals from the outer interval. Thus, without additional assumptions, interpolated intervals do not expand the set of available guaranteed coverage probabilities.

The problems of interval estimating the mean, quantiles, and survival probability in a two-parameter exponential distribution are addressed. Distribution function of a pivotal quantity whose percen...

We construct new pivotals to obtain confidence bounds and confidence intervals for the mean of a stationary process. These follow the approach based on estimating functions. The new pivotals are compared with the standard pivotal based on studentization. We study the first four cumulants of each of these pivotals and explain why the pivotals based on the estimating function approach result in better coverage probabilities. Some simulation results comparing these pivotals have been reported.

Regression analysis is one of the important tools in statistics to investigate the relationships among variables. When the sample size is small, however, the assumptions for regression anal...

13 C Metabolic Fluxes Analysis (13 C MFA) remains to be the most powerful approach to determine intracellular metabolic reaction rates. Decisions on strain engineering and experimentation heavily rely upon the certainty with which these fluxes are estimated. For uncertainty quantification, the vast majority of 13 C MFA studies relies on confidence intervals from the paradigm of Frequentist statistics. However, it is well known that the confidence intervals for a given experimental outcome are not uniquely defined. As a result, confidence intervals produced by different methods can be different, but nevertheless equally valid. This is of high relevance to 13 C MFA, since practitioners regularly use three different approximate approaches for calculating confidence intervals. By means of a computational study with a realistic model of the central carbon metabolism of E. coli, we provide strong evidence that confidence intervals used in the field depend strongly on the technique with which they were calculated and, thus, their use leads to misinterpretation of the flux uncertainty. In order to provide a better alternative to confidence intervals in 13 C MFA, we demonstrate that credible intervals from the paradigm of Bayesian statistics give more reliable flux uncertainty quantifications which can be readily computed with high accuracy using Markov chain Monte Carlo. In addition, the widely applied chi-square test, as a means of testing whether the model reproduces the data, is examined closer.

This paper shows that HAC standard errors must be adjusted when constructing confidence intervals in regressions involving both the factors and idiosyncratic components estimated from a big dataset. This result is in contrast to the seminal result of Bai and Ng (2006) where the assumption that √T/N→0 is sufficient to eliminate the effect of estimation error, where T and N are the time-series and cross-sectional dimensions. Simulations show vast improvements in the coverage rates of the adjusted confidence intervals over the unadjusted ones.

On the basis of failure times of a sample of random size N of iid continuous random variables, we consider the estimation problem of population quantiles of the same distribution. Based on order statistics, confidence intervals for quantile intervals are introduced. Confidence intervals for the difference of quantiles are also investigated. Exact expressions for the coverage probabilities of these intervals are derived and computed numerically. A biometric data set representing the duration of remission of 20 Leukemia patients is used to illustrate the results developed here.

In this paper, we propose empirical likelihood methods based on influence function and jackknife techniques for constructing confidence intervals for mean medical cost with censored data. We conduct a simulation study to compare the coverage probabilities and interval lengths of our proposed confidence intervals with that of the existing normal approximation-based confidence intervals and bootstrap confidence intervals. The proposed methods have better finite-sample performances than existing methods. Finally, we illustrate our proposed methods with a relevant example.

Exact confidence intervals and regions are proposed for the location and scale parameters of the Rayleigh distribution. These sets are valid for complete data, and also for standard and progressive failure censoring. Constrained optimization problems are solved to find the minimum-size confidence sets for the Rayleigh parameters with the required confidence level. The smallest-area confidence region is derived by simultaneously solving four nonlinear equations. Three numerical examples regarding remission times of leukemia patients, strength data and the titanium content in an aircraft-grade alloy, as well as a simulation study, are included for illustrative purposes. Further applications in hypothesis testing and the construction of pointwise and simultaneous confidence bands are also pointed out.

In this work, we propose several different confidence interval methods based on ranked-set samples. First, we develop bootstrap bias-corrected and accelerated method for constructing confidence intervals based on ranked-set samples. Usually, for this method, the accelerated constant is computed by employing jackknife method. Here, we derive an analytical expression for the accelerated constant, which results in reducing the computational burden of this bias-corrected and accelerated bootstrap method. The other proposed confidence interval approaches are based on a monotone transformation along with normal approximation. We also study the asymptotic properties of the proposed methods. The performances of these methods are then compared with those of the conventional methods. Through this empirical study, it is shown that the proposed confidence intervals can be successfully applied in practice. The usefulness of the proposed methods is further illustrated by analyzing a real-life data on shrubs.

One of the indicators for evaluating the capability of a process is the process capability index. In this article bootstrap confidence intervals of generalized process capability index (GPCI) proposed by Maiti et al. (2010) are studied through simulation when the underlying distributions are Lindley and Power Lindley distributions. 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 boot...

In this paper we consider the estimation of R = P[Y < X], where both the random variables X and Y follow Maxwell distributions with different scale parameters. We obtain the maximum likelihood estimator of R. We also provide asymptotic confidence intervals and bootstrap intervals for the same. The Bayes estimator of R is derived under square error loss function and Bayesian credible and HPD intervals are also obtained. The results are illustrated through simulation study and analysis of a real data set.

The well-known Wilson and Agresti–Coull confidence intervals for a binomial proportion p are centered around a Bayesian estimator. Using this as a starting point, similarities between frequentist confidence intervals for proportions and Bayesian credible intervals based on low-informative priors are studied using asymptotic expansions. A Bayesian motivation for a large class of frequentist confidence intervals is provided. It is shown that the likelihood ratio interval for p approximates a Bayesian credible interval based on Kerman’s neutral noninformative conjugate prior up to O(n− 1) in the confidence bounds. For the significance level α ≲ 0.317, the Bayesian interval based on the Jeffreys’ prior is then shown to be a compromise between the likelihood ratio and Wilson intervals. Supplementary materials for this article are available online.

This article deals with constructing confidence intervals/bands for a distribution function based on censored ranked set samples. Toward this end, a resampling plan is suggested and its validity is investigated. Monte Carlo simulations are used to compare performances of the bootstrap confidence intervals with their asymptotic analogs, and their modifications by jackknife. An environmental data set is finally analyzed.

Measuring the difference between two Gini indexes is important in comparing the inequality between two groups. This paper systematically provides confidence intervals for this difference. First, normal approximation- and bootstrap-based confidence intervals are provided for the difference. Second, empirical likelihood (EL)-based confidence intervals are proposed according to a ratio statistic that is proven to have a weighted Chi-squared distribution with one degree of freedom. Third, two calibration approaches are established to improve the undercoverage issue of the EL method; these two approaches are augmented EL and bootstrap-calibrated EL. Monte Carlo simulations show that bootstrap-calibrated EL generally outperforms other methods in constructing confidence intervals. In the simulations, these methods are compared with the permutation method to test the equality of two Gini indexes. Lastly, these methods are applied to real data.