A generalization of the classical concordance correlation coefficient (CCC) is considered under a three-level design where multiple raters rate every subject over time, and each rater is rating every subject multiple times at each measuring time point. The ratings can be discrete or continuous. A methodology is developed for the interval estimation of the CCC based on a suitable linearization of the model along with an adaptation of the fiducial inference approach. The resulting confidence intervals have satisfactory coverage probabilities and shorter expected widths compared to the interval based on Fisher's Z-transformation, even under moderate sample sizes. Two real applications available in the literature are discussed. The first application is based on a clinical trial to determine if various treatments are more effective than a placebo for treating knee pain associated with osteoarthritis. The CCC was used to assess agreement among the manual measurements of the joint space widths on plain radiographs by two raters, and the computer-generated measurements of digitalized radiographs. The second example is on a corticospinal tractography and the CCC was once again applied in order to evaluate the agreement between a well-trained technologist and a neuroradiologist regarding the measurements of fiber number in both the right and left corticospinal tracts. Other relevant applications of our general approach are highlighted in many areas including artificial intelligence.

Generalized fiducial inference on differentiable manifolds

To address the challenge of constructing valid confidence intervals (CIs) for Poisson means in biomedical low-count experiments (e.g., radiation or molecular counting) with known background signals, where existing methods yield overly conservative intervals due to constraints in parameter space. We propose a fiducial framework that redefines the fiducial distribution by adjusting for conditional probability within the restricted parameter space. This computationally efficient approach eliminates empty intervals and leverages parameter constraints to ensure frequentist validity. Numerical simulations demonstrate that the proposed CIs are narrower than conventional methods while maintaining nominal coverage probabilities, particularly near boundary conditions. The method was validated using three real-world biomedical/physics datasets. The fiducial approach provides a robust, statistically efficient solution for Poisson mean inference in restricted spaces. It offers improved precision without compromising coverage, making it highly suitable for analyzing low-count data in biomedical and physical sciences.

Assessing the lifetime of products or materials effectively is essential for formulating maintenance strategies and warranty policies. With advancements in engineering technology, high-reliability, long-lifetime products are becoming increasingly common. To quickly obtain lifetime data for such products, constant-stress accelerated life tests (CSALT) have been widely employed. The failure-free life (FFL) characterizes the early stage during which these products experience no failures, and the three-parameter Weibull distribution (TPWD) models FFL through its threshold parameter. However, the effect of acceleration on the threshold parameter has not been adequately investigated, particularly in terms of modeling and estimation. To solve the above problem, firstly, we introduce a modeling approach based on the linear cumulative exposure model and fatigue theory, which accounts for the acceleration of the threshold parameter in CSALT. Applied to TPWD, the resulting model is referred to as CSALT-TPWD. Then, we examine the non-regular problem in the maximum likelihood estimation of CSALT-TPWD. Next, generalized fiducial inference is employed to provide both point and interval estimates. Furthermore, to enhance the sampling efficiency of the posterior distribution, a Hadamard-based <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell ^{2}$</tex-math></inline-formula> norm prior is proposed. Theoretical analysis confirms that this prior accelerates computation compared to other commonly used norms, while preserving the value of the traditional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell ^{2}$</tex-math></inline-formula> norm prior. Finally, numerical simulations and real case analysis demonstrate that the proposed method captures the characteristics of FFL effectively.

Random-effects meta-analyses with only a few studies often face challenges in accurately estimating between-study heterogeneity, leading to biased effect estimates and confidence intervals with poor coverage. This issue is especially the case when dealing with rare diseases. To address this problem for normally distributed outcomes, two new approaches have been proposed to provide confidence limits of the global mean: one based on fiducial inference, and the other involving two modifications of the signed log-likelihood ratio test statistic in order to have improved performance with small numbers of studies. The performance of the proposed methods was evaluated numerically and compared with the Hartung–Knapp–Sidik–Jonkman approach and its modification to handle small numbers of studies. The simulation results indicated that the proposed methods achieved coverage probabilities closer to the nominal level and produced shorter confidence intervals compared to those based on existing methods. Two real examples are used to illustrate the proposed methods.

Kriging models are widely employed due to their simplicity and flexibility in a variety of fields. To gain more distributional information about the unknown parameters, we focus on constructing the fiducial distribution of Kriging model parameters. To solve the challenge of constructing the fiducial marginal distribution for the spatially related parameter, we substitute the Bayesian posterior distribution for the fiducial distribution of this spatially related parameter and present a quasi-fiducial distribution for Kriging model parameters. A Gibbs sampling algorithm is given to get the samples of the quasi-fiducial distribution. Then a model selection criterion based on the quasi-fiducial distribution is proposed. Numerical studies demonstrate that the proposed method is superior to the Lasso and Elastic Net.

Management research mainly relies on frequentist inference to develop and test theory. Yet, these approaches are limited by their dependence on null hypothesis significance testing, arbitrary p-values, and assumptions about infinite repeated samples. Bayesian techniques address some of these issues by directly modeling probability distributions of parameter estimates, but their reliance on subjective priors can limit theory-building, especially when extant evidence is scarce. Recognizing this need for an inferential technique that combines the accessibility of frequentist inference with the interpretive richness of Bayesian methods, we introduce Generalized Fiducial Inference (GFI) to management research. Grounded in Fisher’s fiducial argument, GFI constructs parameter distributions directly from observed data without requiring priors. As such, GFI provides both the clarity and familiarity of frequentist approaches, enabling researchers to assess whether relationships exist and the probabilistic nuance of Bayesian inference. This allows scholars to understand the likely distribution of these relationships. By removing the necessity of subjective priors, GFI facilitates theory development, improves interpretability, and encourages cumulative knowledge advancement. To demonstrate GFI’s utility, we present simulation studies and an empirical application to showcase its empirical application. Our findings and discussions highlight how GFI can enrich theoretical inferences and foster more robust, meaningful, and cumulative theory-building in management research.

Individual treatment effect estimation has gained significant attention in recent data science literature. This work introduces the Double Neural Network (Double-NN) method to address this problem within the framework of extended fiducial inference (EFI). In the proposed method, deep neural networks are used to model the treatment and control effect functions, while an additional neural network is employed to estimate their parameters. The universal approximation capability of deep neural networks ensures the broad applicability of this method. Numerical results highlight the superior performance of the proposed Double-NN method compared to the conformal quantile regression (CQR) method in individual treatment effect estimation. From the perspective of statistical inference, this work advances the theory and methodology for statistical inference of large models. Specifically, it is theoretically proven that the proposed method permits the model size to increase with the sample size n at a rate of O(nζ) for some 0≤ζ<1, while still maintaining proper quantification of uncertainty in the model parameters. This result marks a significant improvement compared to the range 0≤ζ<12 required by the classical central limit theorem. Furthermore, this work provides a rigorous framework for quantifying the uncertainty of deep neural networks under the neural scaling law, representing a substantial contribution to the statistical understanding of large-scale neural network models.

Post-data statistical inference concerns making probability statements about model parameters conditional on observed data. When a priori knowledge about parameters is available, post-data inference can be conveniently made from Bayesian posteriors. In the absence of prior information, we may still rely on objective Bayes or generalized fiducial inference (GFI). Inspired by approximate Bayesian computation, we propose a novel characterization of post-data inference with the aid of differential geometry. Under suitable smoothness conditions, we establish that Bayesian posteriors and generalized fiducial distributions (GFDs) can be respectively characterized by absolutely continuous distributions supported on the same differentiable manifold: The manifold is uniquely determined by the observed data and the data generating equation of the fitted model. Our geometric analysis not only sheds light on the connection and distinction between Bayesian inference and GFI, but also allows us to sample from posteriors and GFDs using manifold Markov chain Monte Carlo algorithms. A repeated measures analysis of variance example is presented to illustrate the sampling procedure.

In the realm of pairwise lifetime model discrimination, it is a customary practice to frame it as a hypothesis test. In literature, generalized pivotal quantity (GPQ) emerges as an effective tool with complete observations, primarily owing to its advantages in addressing challenges posed by intricate parameter functions and limited sample size. In practical lifetime tests, the occurrence of censoring observations is not uncommon. Under this circumstance, the GPQ-based discrimination is infrequently employed primarily due to the inherent challenge of directly constructing the requisite GPQ. To tackle it, the present study first introduces an algorithm directly integrating data filling with GPQ. Then to mitigate the impact of data filling to GPQ, the generated samples from fiducial distribution also emulate the censoring and filling processes. This novel algorithm is thus designated as the “Mimic filling Algorithm.” For application purposes, this algorithm is applied to Type I censoring data, with the simulation study centered around widely encountered discrimination scenarios for Lognormal, Gamma, and Weibull distributions. In terms of two types errors, simulation results unequivocally demonstrate its superior performance compared to the direct integration of data filling with bootstrap, asymptotic normal approximation, and GPQ. Finally, this study applies the mimic-filling algorithm to discriminate two lithium-ion battery lifetime models with close-fitting results.

The concept of fiducial inference was introduced by R. A. Fisher in the 1930s to address the perceived limitations of Bayesian inference, particularly the need for subjective prior distributions in cases with limited prior information. However, Fisher’s fiducial approach lost favor due to complications, especially in multi-parameter problems. With renewed interest in fiducial inference in the 2000s, generalized fiducial inference (GFI) emerged as a promising extension of Fisher’s ideas, offering new solutions for complex inference challenges. Despite its potential, GFI’s adoption has been hindered by demanding mathematical derivations and complex implementation requirements, such as Markov chain Monte Carlo (MCMC) algorithms. This article introduces AutoGFI, a streamlined variant of GFI designed to simplify its application across various inference problems with additive noise. AutoGFI’s accessibility lies in its simplicity—requiring only a fitting routine—making it a feasible option for a wider range of researchers and practitioners. To demonstrate its efficacy, AutoGFI is applied to three challenging problems: tensor regression, matrix completion, and network cohesion regression. These case studies showcase AutoGFI’s competitive performance against specialized solutions, highlighting its potential to broaden the application of GFI in practical domains, ultimately enriching the statistical inference toolkit.

We develop a novel framework for fiducial inference in linear mixed-effects (LME) models, with the standard deviation of random effects reformulated as coefficients. The exact fiducial density is derived as the equilibrium measure of a reversible Markov chain over the parameter space. The density is equivalent in form to a Bayesian LME with noninformative prior, while the underlying fiducial structure adds new benefits to unify the inference of random effects and all other parameters in a neat and simultaneous way. Our fiducial LME needs no additional tests or statistics for zero variance and is more suitable for small sample sizes. In simulation and empirical analysis, our confidence intervals (CIs) are comparable to those based on Bayesian and likelihood profiling methods. And our inference for the variance of random effects has competitive power with the likelihood ratio test.

Construction of confidence intervals (CIs) for the difference between means and for the ratio of means of two Maxwell distributions is considered. CIs based on the fiducial approach and parametric bootstrap method are proposed and compared with respect to coverage probability and precision. These methods are also extended to find CIs for a difference between percentiles and for a ratio of percentiles of two Maxwell distributions. In particular, CIs for the ratio of the 5th percentiles and for the ratio of the medians are developed and evaluated for accuracy in terms of coverage probability and compared in terms of precision. The methods are illustrated using two examples with real life data.

The objective of this work is to propose the iteratively reweighted least squares concept to form a fiducial generalized pivotal quantity of the between-group variance component for the unbalanced variance components model. The fiducial generalized pivotal quantity is a subclass of the generalized pivotal quantity which is useful technique to deal with problem of nuisance parameters for finding interval estimator. This research provides the probability distribution and the properties of the statistics to lead the constructing of the confidence interval. The authors also prove the construction of the fiducial generalized pivotal quantity through iteratively reweighted least squares. The performance comparison for the new proposed method with other competing methods in the literature is studied through a simulation study. The results of the simulation study demonstrate that the proposed method is very satisfactory in terms of both the coverage probability and the average width of the confidence interval. Furthermore, the analysis of real data for patients of sickle cell disease also illustrates that the proposed method gives the smallest average width of the confidence interval. All these results confirm that the iteratively reweighted least squares fiducial generalized pivotal quantity confidence interval is recommended.

Nozer D. Singpurwalla (1939–2022) We are honored to be guest editors for this special issue of Applied Stochastic Models in Business and Industry which is a tribute to Nozer D. Singpurwalla's scholarly work and achievements. The special issue contains seventeen papers. Four of these papers were based on presented talks at the 2-day conference entitled New Frontiers in Reliability and Risk Analysis, held on October 13–14, 2023 at The George Washington University in Washington, DC. The conference, which was dedicated to Nozer, brought together leading experts and young researchers in the fields in which Nozer was a major contributor, that is reliability, risk analysis, and Bayesian statistics. The special issue includes contributions on these topics from Nozer's friends and colleagues as well as from other researchers. The first article by Soyer and Spizzichino presents an overview of Nozer's work in reliability and risk analysis as well as his interests in foundational aspects of statistics, probability, and decision analysis. The paper by Li, Tierney, Hellmayr, and West deals with sequential Bayesian analysis of multivariate time series models with a focus on causal inference which were both areas of interest to Nozer. The next two papers are on topics that attracted Nozer's attention due to their foundational implications. Sellers and Booker describe their collaborations with Nozer regarding the connections of fuzzy sets with probability and reliability theory. The authors further discuss subsequent advances in this space and the perceptions across disciplines (particularly among statisticians and data scientists) over the last 20 years. The article by Polson and Sokolov presents an introduction to the notions of negative probability, which was of interest to Nozer during his final years, and the authors give a version of Bayes rule for such probabilities. The article by Arkadani, Asadi, and Soofi builds on earlier work by Nozer on the comparison of informativeness of failures versus survivals in life testing. The authors consider a comparison of the information on moments and the model parameters and develop information measures. Finkelstein and Cha present an overview of mixture failure rates (that Nozer often referred to as “predictive failure rate”) to model heterogeneity in reliability and discuss recent developments on the topic including the stochastic intensity paradox. The articles by Limnios, and Palayangoda and Balakrishnan deal with gamma processes for degradation modeling. Nozer used the gamma process in his study of Bayesian life testing, and failure processes in dynamic and multiple failure mode environments. Limnios considers a gamma process for degradation under a random environment modeled by a Markov process and presents results for averaging and normal deviation. Palayangoda and Balakrishnan consider a complete likelihood for the gamma processes and develop inference using the EM algorithm. Lindqvist and Taraldsen present a data-generating function-based approach for simulating exact confidence intervals for reliability and discuss the connection with fiducial inference. Equivalence results are presented for confidence bands for fatigue-life and fatigue-strength models in the article by Liu, Hong, Escobar, and Meeker. The equivalence results are shown for quantile and cumulative distribution functions. Kim and Wilson consider reliability demonstration tests and present a Bayesian approach to identify a set of binomial test plans by taking into account posterior consumer and producer risks. The next two articles deal with system reliability modeling. Lei and Kuo utilize order statistics associated with unit failure times to simplify and make more efficient system reliability calculations. Joint probability distributions for multi-state series and parallel systems with independent components are obtained in Kulkarni, Sabnis, and Ghosh and they are compared with the probability functions from the respective binary versions. Motivated by queueing, Sethuraman considers systems that are subject to interruptions, such as power outages, that cause re-starting of the service provided by the system. Asymptotic results are obtained for a stochastic process, with independent increments, based on “time to complete a service.” Stochastic ordering properties and identifiability issues in latent activation failure models are discussed by Jiang and Basu who consider latent fixed order statistics models as well as hierarchical activation models. The next paper is by Cui, Li, Wan, and Zhang who use model averaging and a jackknife-based weight selection criterion to estimate the conditional average treatment effect in binary response models. The final article by Misaii et al. presents a comparison of the predictive performance of statistical and AI/ML models in the analysis of degradation data and provides insights on different modeling approaches using a case study. This special issue not only serves to honor Nozer's contributions and legacy—it further advances research in the fields of reliability, risk, and Bayesian analysis. The authors declare no conflicts of interest.

Fiducial Inference for Random-Effects Calibration Models: Advancing Reliable Quantification in Environmental Analytical Chemistry

This paper studies the problem of constructing confidence intervals (CIs) for the difference between coefficients of variation (CV) of two censored zero-inflated gamma distributions. Firstly, we propose a Fiducial inference based method that extends the traditional CI construction method of gamma distribution to the scenario of zero-inflated gamma distributions with censored data. Secondly, we propose a Box-Cox transformation based method, in which not only the sample data needs to be transformed, but also the detection limit of gamma distribution with censored data accordingly. Thirdly, we investigate the Method of Variance Estimate Recovery (MOVER), to combine the previous two gamma distribution CI estimates with three binomial distribution CI estimates, and obtain six combination methods. Furthermore, we conduct Monte Carlo simulations to evaluate the performances of the proposed methods, the results indicate that all CI construction methods achieve satisfactory performances in terms of coverage probability, average length and tail error rates. Finally, we perform real data analysis using 11 years of precipitation data from Zhengzhou and Lhasa.

Polycystic ovary syndrome (PCOS) is defined as a chronic low-grade inflammatory reproductive endocrine disorder. PCOS can induce various metabolic disorders, which are associated with a state of mild and slow-acting inflammation. Nevertheless, the causal relationship between polycystic ovary syndrome and inflammatory factors is uncertain. The causality between inflammatory cytokines and PCOS was analyzed by bidirectional Mendelian randomization (MR) in this current probe. We performed an interactive MR study to assess the causal relationships between 91 inflammatory cytokines and PCOS using Genome Wide Association Study (GWAS) data. We underwent dual-sample MR analysis with inverse variance weights (IVW) as the predominant MR methodology with multiple validity and heterogeneity analyses. MR-Egger, weighted median, simple mode, weighted mode and MR-PRESSO were analyzed as multiple likelihood sensitivity analyses to enhance the final results.The results came out interleukin-1-alpha (IL-1A) levels (odds ratio [OR] = 1.051, 95% fiducial interval [95% CI] = 1.009-1.095, P = 0.02) and oncostatin-M (OSM) levels ( [OR] = 1.041, [95% CI] = 1.001-1.082, P = 0.04) were positively associated with the development of PCOS. Moreover, interleukin-7 (IL-7) levels ([OR] = 0.935, [95% CI] = 0.884-0.989, P = 0.02); interleukin-15 receptor subunit alpha (IL15RA) levels ([OR] = 0.959, [95% CI] = 0.929-0.99, P = 0.01); and C-X-C motif chemokine 11 (CXCL11) levels ([OR] = 0.959, [95% CI] = 0.922-0.996. P = 0.03) were strongly negatively associated with PCOS. However, we did not find any strong positive results in the reverse analysis, suggesting that although inflammatory factors contribute to the pathogenesis of PCOS, PCOS itself does not trigger inflammatory factor production.Our study provides genetic evidence for the connection between systemic inflammatory regulators and PCOS. Treatments targeting specific inflammatory factors may help to mitigate the risk of PCOS. The levels of five of the 91 inflammatory factors included in this study, namely, IL1A and OSM, were associated with PCOS. IL1A and OSM contribute to the progression of PCOS while IL-7, IL15RA, and CXCL11 levels are negatively correlated with the development of PCOS.

ABSTRACTA method for constructing exact simulated confidence intervals is presented, valid for situations with both discrete and continuous observations. The idea of the method is to invert a data generating function, which needs not be monotone, and where special attention is taken when the data generating function contains jumps. The method is applied to obtain exact confidence intervals for certain types of censored data from exponential distributions. The censoring schemes under study are earlier treated in the literature, and a comparison to these approaches is considered. The connection to fiducial inference is discussed, and a difference in the paradigm of obtaining intervals for the parameter is studied.

Generalized extreme value (GEV) distribution is used to analyse the maximum from a block of data. It is very useful to describe the unusual event rather than the usual event. In this paper, we propose a change-point detection procedure for GEV distribution based on generalized fiducial inference. The fiducial distribution of the change-point location is constructed. Meanwhile, Markov Chain Monte Carlo method combined with Gibbs sampling and the Metropolis–Hastings algorithm is utilised to estimate the location of the change point and its confidence interval. In addition, the generalized fiducial factor is used to test whether there is a change point. Simulation results show that the proposed generalized fiducial method performs better in accuracy, robustness and the length of confidence intervals. Finally, we apply the proposed method to the annual maximum rainfall data in Beijing.