Mathematics

Research

18 pages, 418 KiB

Open AccessArticle

Inference with Pólya-Gamma Augmentation for US Election Law

by Adam C. Hall and Joseph Kang

Mathematics 2025, 13(6), 945; https://doi.org/10.3390/math13060945 - 13 Mar 2025

Viewed by 80

Pólya-gamma (PG) augmentation has proven to be highly effective for Bayesian MCMC simulation, particularly for models with binomial likelihoods. This data augmentation strategy offers two key advantages. First, the method circumvents the need for analytic approximations or Metropolis–Hastings algorithms, which leads to simpler [...] Read more.

Pólya-gamma (PG) augmentation has proven to be highly effective for Bayesian MCMC simulation, particularly for models with binomial likelihoods. This data augmentation strategy offers two key advantages. First, the method circumvents the need for analytic approximations or Metropolis–Hastings algorithms, which leads to simpler and more computationally efficient posterior inference. Second, the approach can be successfully applied to several types of models, including nonlinear mixed-effects models for count data. The effectiveness of PG augmentation has led to its widespread adoption and implementation in statistical software packages, such as version 2.1 of the R package BayesLogit. This success has inspired us to apply this method to the implementation of Section 203 of the Voting Rights Act (VRA), a US law that requires certain jurisdictions to provide non-English voting materials for specific language minority groups (LMGs). In this paper, we show how PG augmentation can be used to fit a Bayesian model that estimates the prevalence of each LMG in each US voting jurisdiction, and that uses a variable selection technique called stochastic search variable selection. We demonstrate that this new model outperforms the previous model used for 2021 VRA data with respect to model diagnostic measures. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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17 pages, 1088 KiB

Open AccessArticle

Bayesian Estimation of the Stress–Strength Parameter for Bivariate Normal Distribution Under an Updated Type-II Hybrid Censoring

by Yu-Jau Lin, Yuhlong Lio and Tzong-Ru Tsai

Mathematics 2025, 13(5), 792; https://doi.org/10.3390/math13050792 - 27 Feb 2025

Viewed by 199

Abstract

To save time and cost for a parameter inference, the type-II hybrid censoring scheme has been broadly applied to collect one-component samples. In the current study, one of the essential parameters for comparing two distributions, that is, the stress–strength probability [...] Read more.

To save time and cost for a parameter inference, the type-II hybrid censoring scheme has been broadly applied to collect one-component samples. In the current study, one of the essential parameters for comparing two distributions, that is, the stress–strength probability

δ = Pr (X < Y)

, is investigated under a new proposed type-II hybrid censoring scheme that generates the type-II hybrid censored two-component sample from the bivariate normal distribution. The difficult issues occurred from extending the one-component type-II hybrid censored sample to a two-component type-II hybrid censored sample are keeping useful information from both components and the establishment of the corresponding likelihood function. To conquer these two drawbacks, the proposed type-II hybrid censoring scheme is addressed as follows. The observed values of the first component, X, of data pairs

(X, Y)

are recorded up to a random time

τ = max {X_{r : n}, T}

, where

X_{r : n}

is the rth ordered statistic among n items with

r < n

as two pre-specified positive integers and T is a pre-determined experimental time. The observed value from the other component variable Y is recorded only if it is the counterpart of X and also observed before time

τ

; otherwise, it is denoted as occurred or not at

τ

. Under the new proposed scheme, the likelihood function of the new bivariate censored data is derived to include the factors of double improper integrals to cover all possible cases without the loss of data information where any component is unobserved. A Monte Carlo Markov chain (MCMC) method is applied to find the Bayesian estimate of the bivariate distribution model parameters and the stress–strength probability,

δ

. An extensive simulation study is conducted to demonstrate the performance of the developed methods. Finally, the proposed methodologies are applied to a type-II hybrid censored sample generated from a bivariate normal distribution. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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30 pages, 1449 KiB

Open AccessArticle

Inference and Optimal Design on Partially Accelerated Life Tests for the Power Half-Logistic Distribution Under Adaptive Type II Progressive Censoring

by Hanan Haj Ahmad and Mahmoud M. El-Awady

Mathematics 2025, 13(3), 394; https://doi.org/10.3390/math13030394 - 25 Jan 2025

Viewed by 219

Abstract

This study explores accelerated life tests to examine the durability of highly reliable products. These tests involve applying higher stress levels, such as increased temperature, voltage, or pressure, that cause early failures. The power half-logistic (PHL) distribution is utilized due to its flexibility [...] Read more.

This study explores accelerated life tests to examine the durability of highly reliable products. These tests involve applying higher stress levels, such as increased temperature, voltage, or pressure, that cause early failures. The power half-logistic (PHL) distribution is utilized due to its flexibility in modeling the probability density and hazard rate functions, effectively representing various data patterns commonly encountered in practical applications. The step stress partially accelerated life testing model is analyzed under an adaptive type II progressive censoring scheme, with samples drawn from the PHL distribution. The maximum likelihood method estimates model parameters and calculates asymptotic confidence intervals. Bayesian estimates are also obtained using Lindley’s approximation and the Markov Chain Monte Carlo (MCMC) method under different loss functions. Additionally, D- and A-optimality criteria are applied to determine the optimal stress-changing time. Simulation studies are conducted to evaluate the performance of the estimation methods and the optimality criteria. Finally, real-world datasets are analyzed to demonstrate the practical usefulness of the proposed model. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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18 pages, 274 KiB

Open AccessArticle

New Class of Estimators for Finite Population Mean Under Stratified Double Phase Sampling with Simulation and Real-Life Application

by Abdulaziz S. Alghamdi and Hleil Alrweili

Mathematics 2025, 13(3), 329; https://doi.org/10.3390/math13030329 - 21 Jan 2025

Viewed by 340

Abstract

Sampling survey data can sometimes contain outlier observations. When the mean estimator becomes skewed due to the presence of extreme values in the sample, results can be biased. The tendency to remove outliers from sample data is common. However, performing such removal can [...] Read more.

Sampling survey data can sometimes contain outlier observations. When the mean estimator becomes skewed due to the presence of extreme values in the sample, results can be biased. The tendency to remove outliers from sample data is common. However, performing such removal can reduce the accuracy of conventional estimating techniques, particularly with regard to the mean square error (MSE). In order to increase population mean estimation accuracy while taking extreme values into consideration, this study presents an enhanced class of estimators. The method uses extreme values from an auxiliary variable as a source of information rather than eliminating these outliers. Using a first-order approximation, the properties of the suggested class of estimators are investigated within the context of a stratified two-phase sampling framework. A simulation research is conducted to examine the practical performance of these estimators in order to validate the theoretical conclusions. To further demonstrate the superiority of the suggested class of estimators for dealing with extreme values, an analysis of three different datasets demonstrates that they consistently provide higher percent relative efficiency (PRE) when compared to existing estimators. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

22 pages, 983 KiB

Open AccessArticle

A Comparative Study of New Ratio-Type Family of Estimators Under Stratified Two-Phase Sampling

by Abdulaziz S. Alghamdi and Hleil Alrweili

Mathematics 2025, 13(3), 327; https://doi.org/10.3390/math13030327 - 21 Jan 2025

Viewed by 356

Abstract

Two-phase sampling is a useful technique for sample surveys, particularly when prior auxiliary data is not accessible. The ranks of the auxiliary variable often coincide with those of the research variable when two variables are correlated. By considering this relationship, we can significantly [...] Read more.

Two-phase sampling is a useful technique for sample surveys, particularly when prior auxiliary data is not accessible. The ranks of the auxiliary variable often coincide with those of the research variable when two variables are correlated. By considering this relationship, we can significantly increase estimator accuracy. In this paper, we use the ranks of the auxiliary variable along with extreme values to estimate the population mean of the study variable. Up to a first-order approximation, we analyze the characteristics of the suggested class of estimators with an emphasis on biases and mean squared errors in stratified two-phase sampling. The theoretical results are verified using different datasets and a simulation study, which demonstrates that the proposed estimators outperform the existing ones in terms of percent relative efficiency. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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20 pages, 4412 KiB

Open AccessArticle

A Simulation Study on Adaptive Assignment Versus Randomizations in Clinical Trials

by Chien-Tai Lin, Yun-Wei Li and Yi-Jun Hong

Mathematics 2025, 13(1), 44; https://doi.org/10.3390/math13010044 - 26 Dec 2024

Viewed by 461

Abstract

This study investigates a sequential clinical trial comparing two treatments with dichotomous outcomes. We evaluate the effectiveness of five adaptive procedures and three randomization methods for assigning patients to different therapies. The primary objective is to identify an optimal treatment allocation policy that [...] Read more.

This study investigates a sequential clinical trial comparing two treatments with dichotomous outcomes. We evaluate the effectiveness of five adaptive procedures and three randomization methods for assigning patients to different therapies. The primary objective is to identify an optimal treatment allocation policy that maximizes the proportion of successful outcomes in a trial. By comparing the performance of adaptive and randomized procedures, this research provides valuable insights for enhancing treatment allocation strategies in clinical trials, ultimately aiming to improve the overall success rates of therapeutic interventions. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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21 pages, 445 KiB

Open AccessArticle

Analysis of Block Adaptive Type-II Progressive Hybrid Censoring with Weibull Distribution

by Kundan Singh, Yogesh Mani Tripathi, Liang Wang and Shuo-Jye Wu

Mathematics 2024, 12(24), 4026; https://doi.org/10.3390/math12244026 - 22 Dec 2024

Viewed by 493

Abstract

The estimation of unknown model parameters and reliability characteristics is considered under a block adaptive progressive hybrid censoring scheme, where data are observed from a Weibull model. This censoring scheme enhances experimental efficiency by conducting experiments across different testing facilities. Point and interval [...] Read more.

The estimation of unknown model parameters and reliability characteristics is considered under a block adaptive progressive hybrid censoring scheme, where data are observed from a Weibull model. This censoring scheme enhances experimental efficiency by conducting experiments across different testing facilities. Point and interval estimates for parameters and reliability assessments are derived using both classical and Bayesian approaches. The existence and uniqueness of maximum likelihood estimates are established. Consequently, reliability performance and differences across different testing facilities are analyzed. In addition, a Metropolis–Hastings sampling algorithm is developed to approximate complex posterior computations. Approximate confidence intervals and highest posterior density credible intervals are obtained for the parametric functions. The performance of all estimators is evaluated through an extensive simulation study, and observations are discussed. A cancer dataset is analyzed to illustrate the findings under the block adaptive censoring scheme. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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19 pages, 1281 KiB

Open AccessArticle

Testing Spherical Symmetry Based on Statistical Representative Points

by Jiajuan Liang, Ping He and Qiong Liu

Mathematics 2024, 12(24), 3939; https://doi.org/10.3390/math12243939 - 14 Dec 2024

Viewed by 493

Abstract

This paper introduces a novel chisquare test for spherical symmetry, utilizing statistical representative points. The proposed representative-point-based chisquare statistic is shown, through a Monte Carlo study, to considerably improve the power performance compared to the traditional equiprobable chisquare test in many high-dimensional cases. [...] Read more.

This paper introduces a novel chisquare test for spherical symmetry, utilizing statistical representative points. The proposed representative-point-based chisquare statistic is shown, through a Monte Carlo study, to considerably improve the power performance compared to the traditional equiprobable chisquare test in many high-dimensional cases. While the test requires relatively large sample sizes to approximate the chisquare distribution, obtaining critical values from existing chisquare tables is simpler compared to many existing tests for spherical symmetry. A real-data application demonstrates the robustness of the proposed method against different choices of representative points. This paper argues that the use of representative points provides a new perspective in high-dimensional goodness-of-fit testing, offering an alternative approach to evaluating spherical symmetry in such contexts. By leveraging the flexibility of choosing the number of representative points, this method ensures more reliable detection of departures from spherical symmetry, especially in high-dimensional datasets. Overall, this research highlights the practical advantages of the proposed approach in statistical analysis, emphasizing its potential as a powerful tool in goodness-of-fit tests within the realm of high-dimensional data. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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17 pages, 2171 KiB

Open AccessArticle

A Ratio Estimator for the Mean Using a Mixture Optional Enhance Trust (MOET) Randomized Response Model

by Sat Gupta, Michael Parker and Sadia Khalil

Mathematics 2024, 12(22), 3617; https://doi.org/10.3390/math12223617 - 20 Nov 2024

Cited by 1 | Viewed by 544

Abstract

When researchers conduct surveys seeking sensitive, socially stigmatized information, respondents, on average, modify their answers to represent themselves favorably. To overcome this issue, researchers may use Randomized Response Technique (RRT) models. Recently, Parker et al. proposed a model that incorporates some of the [...] Read more.

When researchers conduct surveys seeking sensitive, socially stigmatized information, respondents, on average, modify their answers to represent themselves favorably. To overcome this issue, researchers may use Randomized Response Technique (RRT) models. Recently, Parker et al. proposed a model that incorporates some of the most critical recent quantitative RRT advancements—mixture, optionality, and enhanced trust—into a single model, which they called a Mixture Optional Enhanced (MOET) model. We now improve upon the MOET model by incorporating auxiliary information into the analysis. Positively correlated auxiliary information can improve the mean response estimation through use of a ratio estimator. In this study, we propose just such an estimator for the MOET model. Further, we investigate the conditions under which the ratio estimator outperforms the basic MOET estimator proposed by Parker et al. in 2024. We also consider the possibility that the collection of auxiliary information may compromise privacy; and we study the impact of privacy reduction on the overall model performance as assessed by the unified measure (UM) proposed by Gupta et al. in 2018. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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44 pages, 786 KiB

Open AccessArticle

New Statistical Residuals for Regression Models in the Exponential Family: Characterization, Simulation, Computation, and Applications

by Raydonal Ospina, Patrícia L. Espinheira, Leilo A. Arias, Cleber M. Xavier, Víctor Leiva and Cecilia Castro

Mathematics 2024, 12(20), 3196; https://doi.org/10.3390/math12203196 - 12 Oct 2024

Viewed by 1052

Abstract

Residuals are essential in regression analysis for evaluating model adequacy, validating assumptions, and detecting outliers or influential data. While traditional residuals perform well in linear regression, they face limitations in exponential family models, such as those based on the binomial and Poisson distributions, [...] Read more.

Residuals are essential in regression analysis for evaluating model adequacy, validating assumptions, and detecting outliers or influential data. While traditional residuals perform well in linear regression, they face limitations in exponential family models, such as those based on the binomial and Poisson distributions, due to heteroscedasticity and dependence among observations. This article introduces a novel standardized combined residual for linear and nonlinear regression models within the exponential family. By integrating information from both the mean and dispersion sub-models, the new residual provides a unified diagnostic tool that enhances computational efficiency and eliminates the need for projection matrices. Simulation studies and real-world applications demonstrate its advantages in efficiency and interpretability over traditional residuals. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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10 pages, 259 KiB

Open AccessFeature PaperArticle

Modified Cox Models: A Simulation Study on Different Survival Distributions, Censoring Rates, and Sample Sizes

by Iketle Aretha Maharela, Lizelle Fletcher and Ding-Geng Chen

Mathematics 2024, 12(18), 2903; https://doi.org/10.3390/math12182903 - 18 Sep 2024

Viewed by 1130

Abstract

The classical Cox model is the most popular procedure for studying right-censored data in survival analysis. However, it is based on the fundamental assumption of proportional hazards (PH). Modified Cox models, stratified and extended, have been widely employed as solutions when the PH [...] Read more.

The classical Cox model is the most popular procedure for studying right-censored data in survival analysis. However, it is based on the fundamental assumption of proportional hazards (PH). Modified Cox models, stratified and extended, have been widely employed as solutions when the PH assumption is violated. Nevertheless, prior comparisons of the modified Cox models did not employ comprehensive Monte-Carlo simulations to carry out a comparative analysis between the two models. In this paper, we conducted extensive Monte-Carlo simulation to compare the performance of the stratified and extended Cox models under varying censoring rates, sample sizes, and survival distributions. Our results suggest that the models’ performance at varying censoring rates and sample sizes is robust to the distribution of survival times. Thus, their performance under Weibull survival times was comparable to that of exponential survival times. Furthermore, we found that the extended Cox model outperformed other models under every combination of censoring, sample size and survival distribution. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

33 pages, 530 KiB

Open AccessFeature PaperArticle

Imputing Missing Data in One-Shot Devices Using Unsupervised Learning Approach

by Hon Yiu So, Man Ho Ling and Narayanaswamy Balakrishnan

Mathematics 2024, 12(18), 2884; https://doi.org/10.3390/math12182884 - 15 Sep 2024

Viewed by 970

Abstract

One-shot devices are products that can only be used once. Typical one-shot devices include airbags, fire extinguishers, inflatable life vests, ammo, and handheld flares. Most of them are life-saving products and should be highly reliable in an emergency. Quality control of those productions [...] Read more.

One-shot devices are products that can only be used once. Typical one-shot devices include airbags, fire extinguishers, inflatable life vests, ammo, and handheld flares. Most of them are life-saving products and should be highly reliable in an emergency. Quality control of those productions and predicting their reliabilities over time is critically important. To assess the reliability of the products, manufacturers usually test them in controlled conditions rather than user conditions. We may rely on public datasets that reflect their reliability in actual use, but the datasets often come with missing observations. The experimenter may lose information on covariate readings due to human errors. Traditional missing-data-handling methods may not work well in handling one-shot device data as they only contain their survival statuses. In this research, we propose Multiple Imputation with Unsupervised Learning (MIUL) to impute the missing data using Hierarchical Clustering, k-prototype, and density-based spatial clustering of applications with noise (DBSCAN). Our simulation study shows that MIUL algorithms have superior performance. We also illustrate the method using datasets from the Crash Report Sampling System (CRSS) of the National Highway Traffic Safety Administration (NHTSA). Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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18 pages, 1921 KiB

Open AccessArticle

Predicting Pump Inspection Cycles for Oil Wells Based on Stacking Ensemble Models

by Hua Xin, Shiqi Zhang, Yuhlong Lio and Tzong-Ru Tsai

Mathematics 2024, 12(14), 2231; https://doi.org/10.3390/math12142231 - 17 Jul 2024

Viewed by 860

Abstract

Beam pumping is currently the broadly used method for oil extraction worldwide. A pumpjack shutdown can be incurred by failures from the load, corrosion, work intensity, and downhole working environment. In this study, the duration of uninterrupted pumpjack operation is defined as the [...] Read more.

Beam pumping is currently the broadly used method for oil extraction worldwide. A pumpjack shutdown can be incurred by failures from the load, corrosion, work intensity, and downhole working environment. In this study, the duration of uninterrupted pumpjack operation is defined as the pump inspection cycle. Accurate prediction of the pump inspection cycle can extend the lifespan, reduce unexpected pump accidents, and significantly enhance the production efficiency of the pumpjack. To enhance the prediction performance, this study proposes an improved two-layer stacking ensemble model, which combines the power of the random forests, light gradient boosting machine, support vector regression, and Adaptive Boosting approaches, for predicting the pump inspection cycle. A big pump-related oilfield data set is used to demonstrate the proposed two-layer stacking ensemble model can significantly enhance the prediction quality of the pump inspection cycle. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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