The Net Promoter Score (NPS) is a widely used metric for customer loyalty in business. However, the current theoretical gaps in the literature suggest practical refinements for real-world applications. In this simulation study, we use an unbiased estimator of the variance for the sample NPS to examine coverage and width for three different confidence interval methods: Wald, bootstrap t, and adjusted Wald with weights corresponding to four underlying population distribution shapes: extreme (E), left-skewed (LS), triangular (T), and uniform (U). As the sample size increased, all methods approached the nominal 95% coverage rate with an exception for the extreme population; the adjusted Wald method with triangular and uniform weights is particularly robust among the representative population shapes examined. All adjusted Wald methods performed comparably in width, especially at a larger n. The confidence interval width depended on the population shape. Overall, the Wald and bootstrap t methods should be avoided at small sample sizes and are not recommended. Our methods raise awareness of the sampling distribution of the NPS statistic, provide a theoretical basis for an unbiased estimator of the variance, and assess reliable confidence interval construction. These results provide an informed application of NPS and lay the foundation for future methodological development.

Outlier detection is a fundamental component of data preprocessing and quality monitoring across diverse scientific domains, including engineering, biomedical sciences, and finance. While many variables in controlled environments approximate a normal distribution, real-world data, particularly biological, environmental, and epidemiological measures, are frequently characterized by pronounced right-skewness. To address the shortcomings of conventional methods, this study introduces the Dynamic Threshold for Outlier Detection (DTOD), which reframes outlier detection as a concrete operational workflow. The DTOD framework dynamically adjusts detection thresholds based on a functional relationship between skewness and tail morphology. Validation through large-scale simulation experiments across light-, middle-, and high-skewness levels confirms the method’s versatility. The DTOD proves particularly effective at two ends of the spectrum: enhancing sensitivity for detecting subtle anomalies in light-skewed data while serving as a conservative, high-confidence screening tool that controls false positives in high-skewness environments. In real-world application to North American Association of Central Cancer Registries (NAACCR) data, the method successfully identified outliers with abnormally high unknown tumor size rates in colorectal cancer and maintained a low misclassification rate in highly skewed lung cancer data. Ultimately, the DTOD provides a promising, interpretable solution for improving data quality in skewed scenarios.

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Anatomical variants are observed on paired body sides, yet many prevalence studies—particularly those based on osteological collections—report only right- and left-side frequencies without specifying whether findings occur bilaterally in the same individual. In such cases, the individual-level left–right structure is unobserved. Consequently, inference on laterality and bilateralism cannot be based on the reported data alone and must rely on explicit assumptions about within-individual dependence. We study this problem in the context of anatomic prevalence data, although the framework applies more broadly to paired binary outcomes. We parameterize the admissible joint distributions using a feasibility-based dependence index λ, spanning the full range from independence to maximal feasible concordance implied by the marginal prevalences. Within this framework, we examine two complementary estimands: the paired odds ratio for laterality and bilateral prevalence. Analytic results and Monte Carlo simulations show that bilateral prevalence varies linearly and remains stable across the admissible dependence range, whereas the paired odds ratio exhibits intrinsic boundary instability as dependence approaches its feasible maximum due to vanishing discordant counts. Uncertainty-propagation analyses further indicate that laterality inference is robust to moderate misspecification of the dependence assumption. These results demonstrate that unobserved within-subject dependence is a structural inferential issue in paired binary meta-analysis and motivate feasibility-based sensitivity analysis when only marginal data are available.

The continuation ratio model is a crucial tool for analyzing ordinal response data. However, its explanatory power diminishes under high-dimensional settings where the number of covariates p is large. To address this, we introduce, for the first time, the smoothly clipped absolute deviation (SCAD) penalty into the forward continuation ratio model framework. We propose a corresponding penalized likelihood estimation method that performs simultaneous variable selection and parameter estimation and provides an efficient algorithm for its implementation. Numerical simulations demonstrate the favorable properties of the SCAD penalty: it precisely identifies significant variables while more aggressively shrinking the coefficients of irrelevant ones to zero, outperforming alternative penalties like Lasso and elastic net in selection accuracy. Finally, we illustrate the practical utility of our method through an empirical application using data from the Chinese General Social Survey (CGSS).

The two-parameter Birnbaum–Saunders (B-S) distribution is widely applied across various fields due to its favorable statistical properties. This study aims to enhance the efficiency of modified moment estimators for the B-S distribution by systematically incorporating auxiliary non-sample information. To this end, we developed and analyzed a suite of estimation strategies, including restricted estimators, preliminary test estimators, and Stein-type shrinkage estimators. A pretest procedure was formulated to guide the decision on whether to integrate the non-sample information. The relative performance of these estimators was rigorously evaluated through an asymptotic distributional analysis, comparing their asymptotic distributional bias and risk under a sequence of local alternatives. The finite-sample properties were assessed via Monte Carlo simulation studies. The practical utility of the proposed methods is demonstrated through applications to two real-world datasets: failure times for mechanical valves and bone mineral density measurements. Both numerical results and theoretical analysis confirm that the proposed shrinkage-based techniques deliver substantial efficiency gains over conventional estimators.

Receiver operating characteristic (ROC) surfaces provide a natural extension of ROC curves to three-class diagnostic problems. A key summary index is the volume under the surface (VUS), representing the probability that a randomly chosen observation from each of the three ordered groups is correctly classified. A parametric estimation of VUS typically assumes trinormality of the class distributions. However, a formal method for the verification of this composite assumption has not appeared in the literature. Our approach generalizes the two-class AUC-based GOF test of Zou et al. to the three-class setting by exploiting the parallel structure between empirical and trinormal VUS estimators. We propose a global goodness-of-fit (GOF) test for trinormal ROC models based on the difference between empirical and trinormal parametric estimates of the VUS. To improve stability, a probit transformation is applied and a bootstrap procedure is used to estimate the variance of the difference. The resulting test provides a formal diagnostic for assessing the adequacy of trinormal ROC modeling. Simulation studies illustrate the robustness of the assumption via the empirical size and power of the test under various distributional settings, including skewed and multimodal alternatives. The method’s application to COVID-19 antibody level data demonstrates the practical utility of it. Our findings suggest that the proposed GOF test is simple to implement, computationally feasible for moderate sample sizes, and a useful complement to existing ROC surface methodology.

In this article, we establish a constructive nonparametric scheme for monitoring the quality of services provided by a transportation company. The proposed methodology aims at achieving the diligent tracking of the underlying process and the swift detection of any potential malfunctions. The implementation of the new framework requires the construction of appropriate schemes, which follow the set-up of a Shewhart chart and are connected to ranks and multiple run decision criteria. The dispersion and the mean value of the run length distribution for the suggested distribution-free scheme are investigated for the special case k=2. For illustration purposes, a real-data logistics environment is discussed, whereas the proposed approach is applied for improving the quality of the provided services.

In this article, we propose a goodness-of-fit test for a one-parameter Lindley distribution based on energy statistics. The Lindley distribution has been widely used in reliability studies and survival analysis, especially in applied sciences. The proposed test procedure is simple and more powerful against general alternatives. Under different settings, Monte Carlo simulations show that the proposed test is able to be well controlled for any given nominal levels. In terms of power, the proposed test outperforms other existing similar methods in different settings. We then apply the proposed test to real-life datasets to demonstrate its competitiveness and usefulness.

A standard approach for assessing the performance of partition models is to create synthetic datasets with a prespecified clustering structure and assess how well the model reveals this structure. A common format involves subjects being assigned to different clusters, with observations simulated so that subjects within the same cluster have similar profiles, allowing for some variability. In this manuscript, we consider observations from interval variables. Interval data are commonly observed in cohort and Genome-Wide Association studies, and our focus is on Single-Nucleotide Polymorphisms. Theoretical and empirical results are utilized to explore the dependence structure between the variables in relation to the clustering structure for the subjects. A novel algorithm is proposed that allows control over the marginal stratified correlation structure of the variables, specifying exact correlation values within groups of variables. Practical examples are shown, and a synthetic dataset is compared to a real one, to demonstrate similarities and differences.