Constructing Confidence Intervals Research Articles

Conditional parametric bootstrap in GLARMA models

Due to situations where asymptotic results are not valid or when we do not know the true distributions of the data set, the bootstrap technique stands out as an effective alternative to make inferences about model parameters. In this context, we introduce the conditional parametric bootstrap, applied to time series data, specifically using the Generalized Linear Autoregressive Moving Average (GLARMA) model. Our interest is to use the procedure to correct possible biases and construct confidence intervals and hypothesis testing for the parameters of the GLARMA model. We conducted simulation studies to evaluate the performance of the procedure in finite samples. The results show that the proposed technique is promising and can be applied in practical situations. An example is the analysis of the monthly number of chronic obstructive pulmonary disease (COPD) cases, which illustrates the potential of the proposed approach in a real situation.

A Monte Carlo Investigation of Confidence Intervals for a Nondecreasing Series

We consider the problem of estimating a time series of population means from a series of sample surveys when the means are known to be nondecreasing. We introduce the standard survey estimators of the series of means, which are not guaranteed to be nondecreasing. We employ the Pool Adjacent Violators Algorithm (PAVA) to turn the series of standard survey estimates into a nondecreasing series. We introduce five methods of constructing confidence intervals for the series of non-decreasing means: normal-theory intervals based on the standard survey point estimator of the mean and the Taylor series estimator of its variance; normal-theory intervals based on the nondecreasing PAVA estimator of the mean and a unique jackknife estimator of its variance developed here (jackknife-[Formula: see text]); intervals similar to the aforementioned method but based on Student’s-[Formula: see text] distribution (jackknife-[Formula: see text]); analytical intervals due to Morris; and simultaneous confidence limits due to Korn. We report the results of a Monte Carlo simulation and assess the methods’ performance under various scenarios. Jackknife-[Formula: see text] intervals exhibited coverage probabilities that are near the nominal value. Taylor, jackknife-[Formula: see text], and Korn intervals found coverage below the nominal value while Morris intervals were excessively conservative. Jackknife intervals were much narrower than Taylor intervals.

Asymptotic normality of a relative error functional regression estimator under left truncation and right censoring

. In this article, we introduce a non parametric regression estimator based on a relative error criterion, for left truncated and right censored data with functional predictors. Left truncation and right censoring are common forms of incomplete data in survival analysis, where the event time of interest is only observed within a certain interval which adds some challenges to the regression task. We establish the asymptotic normality of our estimator and develop a procedure for constructing confidence intervals for the predicted responses. The theoretical results obtained are supported using simulation studies and a real data example from the energy field.

Code and Data Repository for Simulating Confidence Intervals for Conditional Value-at-Risk via Least-Squares Metamodels

The goal of this software is to demonstrate the effect of constructing confidence intervals for CVaR proposed in "Simulating Confidence Intervals for Conditional Value-at-Risk via Least-Squares Metamodels" by Qidong Lai, Guangwu Liu, Bingfeng Zhang and Kun Zhang.

Speeding up interval estimation for R2-based mediation effect of high-dimensional mediators via cross-fitting.

Mediation analysis is a useful tool in investigating how molecular phenotypes such as gene expression mediate the effect of exposure on health outcomes. However, commonly used mean-based total mediation effect measures may suffer from cancellation of component-wise mediation effects in opposite directions in the presence of high-dimensional omics mediators. To overcome this limitation, we recently proposed a variance-based R-squared total mediation effect measure that relies on the computationally intensive nonparametric bootstrap for confidence interval estimation. In the work described herein, we formulated a more efficient two-stage, cross-fitted estimation procedure for the R2 measure. To avoid potential bias, we performed iterative Sure Independence Screening (iSIS) in two subsamples to exclude the non-mediators, followed by ordinary least squares regressions for the variance estimation. We then constructed confidence intervals based on the newly derived closed-form asymptotic distribution of the R2 measure. Extensive simulation studies demonstrated that this proposed procedure is much more computationally efficient than the resampling-based method, with comparable coverage probability. Furthermore, when applied to the Framingham Heart Study, the proposed method replicated the established finding of gene expression mediating age-related variation in systolic blood pressure and identified the role of gene expression profiles in the relationship between sex and high-density lipoprotein cholesterol level. The proposed estimation procedure is implemented in R package CFR2M.

Simultaneous confidence interval construction for many-to-one of proportion ratios of bilateral correlated data.

In ophthalmology and otolaryngology, data collected from paired body parts are typically reformatted into categorical bilateral data structures for subsequent research. This article applies Donner's equal correlation coefficient model and obtains nine simultaneous confidence intervals (SCI) of proportion ratios under three asymptotic statistical methods and three ways of multiplicity adjustment. The empirical coverage probability and mean interval width are evaluated through Monte Carlo simulations. A real example is used to demonstrate the proposed methods.

Multimedia: Multimodal Mediation Analysis of Microbiome Data.

Mediation analysis has emerged as a versatile tool for answering mechanistic questions in microbiome research because it provides a statistical framework for attributing treatment effects to alternative causal pathways. Using a series of linked regressions, this analysis quantifies how complementary data relate to one another and respond to treatments. Despite these advances, existing software's rigid assumptions often result in users viewing mediation analysis as a black box. We designed the multimedia R package to make advanced mediation analysis techniques accessible, ensuring that statistical components are interpretable and adaptable. The package provides a uniform interface to direct and indirect effect estimation, synthetic null hypothesis testing, bootstrap confidence interval construction, and sensitivity analysis, enabling experimentation with various mediator and outcome models while maintaining a simple overall workflow. The software includes modules for regularized linear, compositional, random forest, hierarchical, and hurdle modeling, making it well-suited to microbiome data. We illustrate the package through two case studies. The first re-analyzes a study of the microbiome and metabolome of Inflammatory Bowel Disease patients, uncovering potential mechanistic interactions between the microbiome and disease-associated metabolites, not found in the original study. The second analyzes new data about the influence of mindfulness practice on the microbiome. The mediation analysis highlights shifts in taxa previously associated with depression that cannot be explained indirectly by diet or sleep behaviors alone. A gallery of examples and further documentation can be found at https://go.wisc.edu/830110.

Statistical inference of entropy functions of generalized inverse exponential model under progressive type-II censoring test.

This article explores the estimation of Shannon entropy and Rényi entropy based on the generalized inverse exponential distribution under the condition of stepwise Type II truncated samples. Firstly, we analyze the maximum likelihood estimation and interval estimation of Shannon entropy and Rényi entropy for the generalized inverse exponential distribution. In this process, we use the bootstrap method to construct confidence intervals for Shannon entropy and Rényi entropy. Next, we select the gamma distribution as the prior distribution and apply the Lindley approximation algorithm to calculate `estimates of Shannon entropy and Rényi entropy under different loss functions including Linex loss function, entropy loss function, and DeGroot loss function respectively. Afterwards, simulation is used to calculate estimates and corresponding mean square errors of Shannon entropy and Rényi entropy in GIED model. The research results show that under DeGroot loss function, estimation accuracy of Shannon entropy and Rényi entropy for generalized inverse exponential distribution is relatively high, overall Bayesian estimation performs better than maximum likelihood estimation. Finally, we demonstrate effectiveness of our estimation method in practical applications using a set of real data.

Speeding up interval estimation for -based mediation effect of high-dimensional mediators via cross-fitting.

Mediation analysis is a useful tool in investigating how molecular phenotypes such as gene expression mediate the effect of exposure on health outcomes. However, commonly used mean-based total mediation effect measures may suffer from cancellation of component-wise mediation effects in opposite directions in the presence of high-dimensional omics mediators. To overcome this limitation, we recently proposed a variance-based R-squared total mediation effect measure that relies on the computationally intensive nonparametric bootstrap for confidence interval estimation. In the work described herein, we formulated a more efficient two-stage, cross-fitted estimation procedure for the measure. To avoid potential bias, we performed iterative Sure Independence Screening (iSIS) in two subsamples to exclude the non-mediators, followed by ordinary least squares regressions for the variance estimation. We then constructed confidence intervals based on the newly derived closed-form asymptotic distribution of the measure. Extensive simulation studies demonstrated that this proposed procedure is much more computationally efficient than the resampling-based method, with comparable coverage probability. Furthermore, when applied to the Framingham Heart Study, the proposed method replicated the established finding of gene expression mediating age-related variation in systolic blood pressure and identified the role of gene expression profiles in the relationship between sex and high-density lipoprotein cholesterol level. The proposed estimation procedure is implemented in R package CFR2M.

Global Inference and Test for Eigensystems of Imaging Data Over Complicated Domains

A nonparametric approach for analyzing eigensystems of image data over a complex domain is novelly developed. The proposed estimators, which are based on bivariate splines, have both oracle efficiency of O p ( n − 1 / 2 ) , meaning they are asymptotically indistinguishable from estimators computed with true trajectories, and computational efficiency with more convenient spectrum decomposition forms. Under mild conditions, we derive consistency and weak convergence of our proposed estimators, which can be used to construct confidence intervals and simultaneous confidence corridors for any individual eigenvalue and eigenfunction, as well as design uniform inference procedures for eigensystems with a diverging number of components. In addition, we extend our method to two-sample test problems. Extensive simulation results provide strong support for the asymptotic theory, and the procedures are applied to two potential seawater temperature data sets to demonstrate its validity.

Simultaneous Confidence Intervals for Signal Detection and Ascertaining Precision of Adverse Event Rates in Clinical Trials

The marketing authorization of a medicinal product is contingent upon demonstration of safety and efficacy in support of the product’s labeled conditions of use. To demonstrate safety, one group of adverse events that requires detailed consideration is common adverse reactions (ARs). ARs and their frequency are reported in prescription drug labeling in the US and EU. The determination of these adverse reactions generally takes a simple approach—usually, inclusion is through the frequency of reporting and whether the adverse event (AE) rate for the drug exceeds the placebo rate. This standard method does not account for confounders or multiplicity. To overcome these limitations, we propose a Monte Carlo approach to detect drug safety signals in clinical trials. We fit regression models incorporating covariates to assess the drug effect on the rate of AEs. Adjustment for multiplicity is carried out through the construction of simultaneous confidence intervals accounting for arbitrary correlations. A computationally efficient multiplier bootstrap approach using the Rademacher sequences is developed to generate random samples from the joint distribution of the estimators for all the AE rates. Compared to Bonferroni-based methods, the proposed method leads to narrower simultaneous confidence intervals and is more powerful in detecting potential safety signals.

INFERENCE IN MILDLY EXPLOSIVE AUTOREGRESSIONS UNDER UNCONDITIONAL HETEROSKEDASTICITY

Mildly explosive autoregressions have been extensively employed in recent theoretical and applied econometric work to model the phenomenon of asset market bubbles. An important issue in this context concerns the construction of confidence intervals for the autoregressive parameter that represents the degree of explosiveness. Existing studies rely on intervals that are justified only under conditional homoskedasticity/heteroskedasticity. This paper studies the problem of constructing asymptotically valid confidence intervals in a mildly explosive autoregression where the innovations are allowed to be unconditionally heteroskedastic. The assumed variance process is general and can accommodate both deterministic and stochastic volatility specifications commonly adopted in the literature. Within this framework, we show that the standard heteroskedasticity- and autocorrelation-consistent estimate of the long-run variance converges in distribution to a nonstandard random variable that depends on nuisance parameters. Notwithstanding this result, the corresponding t-statistic is shown to still possess a standard normal limit distribution. To improve the quality of inference in small samples, we propose a dependent wild bootstrap-t procedure and establish its asymptotic validity under relatively weak conditions. Monte Carlo simulations demonstrate that our recommended approach performs favorably in finite samples relative to existing methods across a wide range of volatility specifications. Applications to international stock price indices and U.S. house prices illustrate the relevance of the advocated method in practice.

Uncertainty quantification in high-dimensional linear models incorporating graphical structures with applications to gene set analysis.

The functions of genes in networks are typically correlated due to their functional connectivity. Variable selection methods have been developed to select important genes associated with a trait while incorporating network graphical information. However, no method has been proposed to quantify the uncertainty of individual genes under such settings. In this paper, we construct confidence intervals (CIs) and provide P-values for parameters of a high-dimensional linear model incorporating graphical structures where the number of variables p diverges with the number of observations. For combining the graphical information, we propose a graph-constrained desparsified LASSO (least absolute shrinkage and selection operator) (GCDL) estimator, which reduces dramatically the influence of high correlation of predictors and enjoys the advantage of faster computation and higher accuracy compared with the desparsified LASSO. Theoretical results show that the GCDL estimator achieves asymptotic normality. The asymptotic property of the uniform convergence is established, with which an explicit expression of the uniform CI can be derived. Extensive numerical results indicate that the GCDL estimator and its (uniform) CI perform well even when predictors are highly correlated. An R package implementing the proposed method is available at https://github.com/XiaoZhangryy/gcdl.

Confidence Intervals for the Mean and Difference of Means of Birnbaum-Saunders Distributions with Application to Wind Speed Data

This paper proposes the confidence intervals for the mean and difference of means of Birnbaum-Saunders (BirSau) distributions based on the Bootstrap confidence interval (BCI), Percentile bootstrap confidence interval (PBCI), Generalized confidence interval (GCI), Bayesian credible interval (BayCrI) and the highest posterior density (HPD). The simulation study used R statistical software to evaluate the coverage probabilities and average lengths. The concerning results of the mean suggest that HPD is the recommended method for constructing confidence intervals in the BirSau distributions, except for small sample sizes where the GCI method proves more efficient. For the difference of means, PBCI emerges as the preferred way to construct confidence intervals, except in some cases where small sample sizes with the HPD method are more efficient. Moreover, the average lengths of these proposed confidence intervals decreased as both sample size and shape parameters increased. To illustrate the effectiveness of the suggested confidence intervals, we applied them to wind speed datasets collected in Ayutthaya and Ratchaburi provinces, Thailand.

Uncertainty quantification and confidence intervals for naive rare-event estimators

Abstract We consider the estimation of rare-event probabilities using sample proportions output by naive Monte Carlo or collected data. Unlike using variance reduction techniques, this naive estimator does not have an a priori relative efficiency guarantee. On the other hand, due to the recent surge of sophisticated rare-event problems arising in safety evaluations of intelligent systems, efficiency-guaranteed variance reduction may face implementation challenges which, coupled with the availability of computation or data collection power, motivate the use of such a naive estimator. In this paper we study the uncertainty quantification, namely the construction, coverage validity, and tightness of confidence intervals, for rare-event probabilities using only sample proportions. In addition to the known normality, Wilson, and exact intervals, we investigate and compare them with two new intervals derived from Chernoff’s inequality and the Berry–Esseen theorem. Moreover, we generalize our results to the natural situation where sampling stops by reaching a target number of rare-event hits. Our findings show that the normality and Wilson intervals are not always valid, but they are close to the newly developed valid intervals in terms of half-width. In contrast, the exact interval is conservative, but safely guarantees the attainment of the nominal confidence level. Our new intervals, while being more conservative than the exact interval, provide useful insights into understanding the tightness of the considered intervals.

Likelihood-Ratio-Based Confidence Intervals for Multiple Threshold Parameters

Abstract This paper proposes the inversion of likelihood ratio tests for the construction of confidence intervals for multiple threshold parameters. Using Monte Carlo simulations, conservative likelihood-ratio-based confidence intervals are shown to exhibit empirical coverage rates at least as high as nominal levels for all threshold parameters, while still being informative in the sense of only including relatively few observations in each confidence interval. These findings are robust to the magnitude of the threshold effect, the sample size and the presence of serial correlation. Applications to existing models with multiple thresholds for U.S. real GDP growth and for the wage Phillips curve demonstrate how the proposed approach is empirically relevant to make inferences about the uncertainty of threshold estimates.

Analysis of the lifetime performance index in Gompertz distribution based on k-record values

The lifetime performance index plays a crucial role in manufacturing procedures as it provides valuable insights into the overall quality of a product. Considering the Gompertz distribution as a widely applicable statistical distribution, this index to emphasize the effect in analyzing lifetime data. This paper investigates the inference techniques associated with the lifetime performance index for the Gompertz distribution when k-record values are available. For this problem, two innovative methods are introduced utilizing the concept of the generalized pivotal variable. These methods are not only applicable to hypothesis tests but also enable the construction of confidence intervals. To further enhance the accuracy of estimation, the study proposes Bayesian estimation techniques based on the Metropolis–Hastings algorithms. To assess the efficacy of these methods, a simulation study is conducted. The results demonstrate that all three procedures exhibit exceptional performance, even when confronted with datasets consisting of small k-record values. Moreover, a dataset is utilized to demonstrate the implementation of these three methods.

Statistical inference on kurtosis of independent component model

Independent component model (ICM) is a widely-used population distribution in high-dimensional data analysis and random matrix theory. In this work, we study the kurtosis of the ICM, which is an important parameter in asymptotic distributions of many commonly used statistics and is also an important criterion for measuring the heavy-tailed nature of the data. Based on U-statistics, we develop an estimation method. Theoretically, we show that the proposed estimator is consistent under regular conditions, especially we relax the restriction that the data dimension and the sample size are of the same order. Furthermore, we derive the asymptotic normality of the estimator, which allows us to construct confidence intervals and hypothesis testing statistics. Computationally, we provide a fast and efficient algorithm by leveraging the matrix structure, where the computational complexity is essentially equivalent to computing the sample covariance matrix and the Gram matrix. Finally, the effectiveness of the proposed method is demonstrated through simulated data and real-world data.

Reparametrized Firth's Logistic Regressions for Dose-Finding Study With the Biased-Coin Design.

Finding an adequate dose of the drug by revealing the dose-response relationship is very crucial and a challenging problem in the clinical development. The main concerns in dose-finding study are to identify a minimum effective dose (MED) in anesthesia studies and maximum tolerated dose (MTD) in oncology clinical trials. For the estimation of MED and MTD, we propose two modifications of Firth's logistic regression using reparametrization, called reparametrized Firth's logistic regression (rFLR) and ridge-penalized reparametrized Firth's logistic regression (RrFLR). The proposed methods are designed by directly reducing the small-sample bias of the maximum likelihood estimate for the parameter of interest. In addition, we develop a method on how to construct confidence intervals for rFLR and RrFLR using profile penalized likelihood. In the up-and-down biased-coin design, numerical studies confirm the superior performance of the proposed methods in terms of the mean squared error, bias, and coverage accuracy of confidence intervals.

Bootstrap-based statistical inference for linear mixed effects under misspecifications

Linear mixed effects are considered excellent predictors of cluster-level parameters in various domains. However, previous research has demonstrated that their performance is affected by departures from model assumptions. Given the common occurrence of these departures in empirical studies, there is a need for inferential methods that are robust to misspecifications while remaining accessible and appealing to practitioners. Statistical tools have been developed for cluster-wise and simultaneous inference for mixed effects under distributional misspecifications, employing a user-friendly semiparametric random effect bootstrap. The merits and limitations of this approach are discussed in the general context of model misspecification. Theoretical analysis demonstrates the asymptotic consistency of the methods under general regularity conditions. Simulations show that the proposed intervals are robust to departures from modelling assumptions, including asymmetry and long tails in the distributions of errors and random effects, outperforming competitors in terms of empirical coverage probability. Finally, the methodology is applied to construct confidence intervals for household income across counties in the Spanish region of Galicia.