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Mathematics
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Reliability Estimation and Mathematical Statistics
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Reliability Estimation and Mathematical Statistics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "D1: Probability and Statistics".

Deadline for manuscript submissions: 31 December 2025 | Viewed by 9419

Special Issue Editor

Prof. Dr. Kamel Rekab

E-Mail Website
Guest Editor
Division of Computing Analytics and Mathematics, School of Science and Engineering, University of Missouri Kansas City, 5110 Rockhill Road, Kansas City, MO 64110, USA
Interests: statistical software testing and reliability; network security; biostatistics; statistics in advanced manufacturing; statistical quality improvement; design of industrial experiments; sequential analysis; mathematical statistics; probability theory

Special Issue Information

Dear Colleagues,

Reliability estimation and mathematical statistics are two closely intertwined fields that play a crucial role in various scientific disciplines and industries. Reliability estimation focuses on assessing the dependability and performance of systems, products, or processes over time. It involves the analysis of failure data to make informed predictions about future reliability. Mathematical statistics, on the other hand, provides the theoretical framework and tools for drawing meaningful conclusions from data, making it an indispensable component of reliability analysis.

In reliability estimation, engineers and statisticians utilize various statistical techniques such as survival analysis, hazard functions, and probability distributions to model and quantify the likelihood of failures or breakdowns. This information is invaluable for decision-making in fields like engineering, manufacturing, healthcare, and finance, where reliability is a critical concern.

Mathematical statistics underpins these reliability assessments by offering methods for data collection, hypothesis testing, and parameter estimation. It involves concepts like sampling theory, probability theory, and statistical inference to extract meaningful insights from empirical data. By applying mathematical statistics, analysts can make informed decisions about system maintenance, quality control, and risk management.

In conclusion, reliability estimation and mathematical statistics are integral components of modern problem-solving and decision-making processes. Their symbiotic relationship empowers industries and researchers to improve the dependability and performance of systems, products, and processes, ultimately leading to safer and more efficient outcomes across a wide range of applications. 

It is with great enthusiasm and anticipation that I write to you today as the Guest Editor of the upcoming Special Issue Reliability Estimation and Mathematical Statistics. Our commitment to excellence and innovation in the field has led us to a momentous juncture, and I invite you all to be a part of this exciting journey.

We extend a warm invitation to contribute your work and insights to Reliability Estimation and Mathematical Statistics. The hope is to offer readers fresh insights and address topics of critical importance in reliability estimation and mathematical statistics.

I encourage you to explore our submission guidelines and consider joining us as contributors, and for our dedicated readers, please continue to engage with us by providing feedback and sharing your thoughts on our content.

Thank you for your continued support, and I look forward to our shared exploration of ideas and insights.

Prof. Dr. Kamel Rekab
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • estimation
  • reliability
  • sequential
  • Bayesian
  • classical
  • decision rule
  • simulation
  • software
  • system

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Published Papers (7 papers)

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15 pages, 1736 KiB  
Article
Mathematical Models of Critical Soft Error in Synchronous and Self-Timed Pipeline
by Igor Sokolov, Yuri Stepchenkov, Yuri Diachenko and Dmitry Khilko
Mathematics 2025, 13(5), 695; https://doi.org/10.3390/math13050695 - 21 Feb 2025
Viewed by 497
Abstract
This paper analyzes the impact of a single soft error on the performance of a synchronous and self-timed pipeline. A nuclear particle running through the integrated circuit body is considered the most probable soft error source. The existing estimates show that self-timed circuits [...] Read more.
This paper analyzes the impact of a single soft error on the performance of a synchronous and self-timed pipeline. A nuclear particle running through the integrated circuit body is considered the most probable soft error source. The existing estimates show that self-timed circuits offer an advantage in terms of single soft error tolerance. The paper proves these estimates on the basis of a comparative probability analysis of a critical fault in two types of pipelines. The mathematical models derived in the paper describe the probability of a critical fault depending on the circuit’s characteristics, its operating discipline, and the soft error parameters. The self-timed pipeline operates in accordance with a two-phase discipline, based on the request–acknowledge interaction within the pipeline’s stages, which provides it with increased immunity to soft errors. Quantitative calculations performed on the basis of the derived mathematical models show that the self-timed pipeline has about 6.1 times better tolerance to a single soft error in comparison to its synchronous counterpart. The obtained results are in good agreement with empirical estimates of the soft error tolerance level of synchronous and self-timed circuits. Full article
(This article belongs to the Special Issue Reliability Estimation and Mathematical Statistics)
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24 pages, 413 KiB  
Article
Bayesian Methods for Step-Stress Accelerated Test under Gamma Distribution with a Useful Reparametrization and an Industrial Data Application
by Hassan S. Bakouch, Fernando A. Moala, Shuhrah Alghamdi and Olayan Albalawi
Mathematics 2024, 12(17), 2747; https://doi.org/10.3390/math12172747 - 4 Sep 2024
Viewed by 1067
Abstract
This paper presents a multiple step-stress accelerated life test using type II censoring. Assuming that the lifetimes of the test item follow the gamma distribution, the maximum likelihood estimation and Bayesian approaches are used to estimate the distribution parameters. In the Bayesian approach, [...] Read more.
This paper presents a multiple step-stress accelerated life test using type II censoring. Assuming that the lifetimes of the test item follow the gamma distribution, the maximum likelihood estimation and Bayesian approaches are used to estimate the distribution parameters. In the Bayesian approach, new parametrizations can lead to new prior distributions and can be a useful technique to improve the efficiency and effectiveness of Bayesian modeling, particularly when dealing with complex or high-dimensional models. Therefore, in this paper, we present two sets of prior distributions for the parameters of the accelerated test where one of them is based on the reparametrization of the other. The performance of the proposed prior distributions and maximum likelihood approach are investigated and compared by examining the summaries and frequentist coverage probabilities of intervals. We introduce the Markov Chain Monte Carlo (MCMC) algorithms to generate samples from the posterior distributions in order to evaluate the estimators and intervals. Numerical simulations are conducted to examine the approach’s performance and one-sample lifetime data are presented to illustrate the proposed methodology. Full article
(This article belongs to the Special Issue Reliability Estimation and Mathematical Statistics)
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26 pages, 3068 KiB  
Article
A New Generalization of the Uniform Distribution: Properties and Applications to Lifetime Data
by Isidro Jesús González-Hernández, Luis Carlos Méndez-González, Rafael Granillo-Macías, José Luis Rodríguez-Muñoz and José Sergio Pacheco-Cedeño
Mathematics 2024, 12(15), 2328; https://doi.org/10.3390/math12152328 - 25 Jul 2024
Cited by 2 | Viewed by 1551
Abstract
In this paper, we generalize two new statistical distributions, to improve the ability to model failure rates with non-monotonic, monotonic, and mainly bathtub curve behaviors. We call these distributions Generalized Powered Uniform Distribution and MOE-Powered Uniform. The proposed distributions’ approach is based on [...] Read more.
In this paper, we generalize two new statistical distributions, to improve the ability to model failure rates with non-monotonic, monotonic, and mainly bathtub curve behaviors. We call these distributions Generalized Powered Uniform Distribution and MOE-Powered Uniform. The proposed distributions’ approach is based on incorporating a parameter k in the power of the values of the random variables, which is associated with the Probability Density Function and includes an operator called the Powered Mean. Various statistical and mathematical features focused on reliability analysis are presented and discussed, to make the models attractive to reliability engineering or medicine specialists. We employed the Maximum Likelihood Estimator method to estimate the model parameters and we analyzed its performance through a Monte Carlo simulation study. To demonstrate the flexibility of the proposed approach, a comparative analysis was carried out on four case studies with the proposed MOE-Powered Uniform distribution, which can model failure times as a bathtub curve. The results showed that this new model is more flexible and useful for performing reliability analysis. Full article
(This article belongs to the Special Issue Reliability Estimation and Mathematical Statistics)
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18 pages, 385 KiB  
Article
New and Efficient Estimators of Reliability Characteristics for a Family of Lifetime Distributions under Progressive Censoring
by Syed Ejaz Ahmed, Reza Arabi Belaghi, Abdulkadir Hussein and Alireza Safariyan
Mathematics 2024, 12(10), 1599; https://doi.org/10.3390/math12101599 - 20 May 2024
Cited by 3 | Viewed by 1163
Abstract
Estimation of reliability and stress–strength parameters is important in the manufacturing industry. In this paper, we develop shrinkage-type estimators for the reliability and stress–strength parameters based on progressively censored data from a rich class of distributions. These new estimators improve the performance of [...] Read more.
Estimation of reliability and stress–strength parameters is important in the manufacturing industry. In this paper, we develop shrinkage-type estimators for the reliability and stress–strength parameters based on progressively censored data from a rich class of distributions. These new estimators improve the performance of the commonly used Maximum Likelihood Estimators (MLEs) by reducing their mean squared errors. We provide analytical asymptotic and bootstrap confidence intervals for the targeted parameters. Through a detailed simulation study, we demonstrate that the new estimators have better performance than the MLEs. Finally, we illustrate the application of the new methods to two industrial data sets, showcasing their practical relevance and effectiveness. Full article
(This article belongs to the Special Issue Reliability Estimation and Mathematical Statistics)
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23 pages, 523 KiB  
Article
A Net Present Value Analysis of Opportunity-Based Age Replacement Models in Discrete Time
by Jing Wu, Cunhua Qian and Tadashi Dohi
Mathematics 2024, 12(10), 1472; https://doi.org/10.3390/math12101472 - 9 May 2024
Viewed by 1282
Abstract
Two important opportunistic age replacement models, under replacement first and last disciplines, are generalized in discrete time. The net present value (NPV) is applied to formulate the expected total costs. The priority of multiple replacement options is considered to classify the cost model [...] Read more.
Two important opportunistic age replacement models, under replacement first and last disciplines, are generalized in discrete time. The net present value (NPV) is applied to formulate the expected total costs. The priority of multiple replacement options is considered to classify the cost model with discounting into six cases. Since the NPV method accurately calculates the expected replacement costs over an infinite horizon in an unstable economic environment, we discuss some optimal opportunistic age replacement policies which minimize the expected total discounted costs over an infinite time horizon. Furthermore, we formulate a unified model under each discipline, merging six discrete time replacement models with probabilistic priority. Finally, a case study on optimal replacement first and last policies for pole air switches in a Japanese power company is presented. Full article
(This article belongs to the Special Issue Reliability Estimation and Mathematical Statistics)
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25 pages, 1072 KiB  
Article
Multivariate Structural Equation Modeling Techniques for Estimating Reliability, Measurement Error, and Subscale Viability When Using Both Composite and Subscale Scores in Practice
by Walter Peter Vispoel, Hyeryung Lee and Tingting Chen
Mathematics 2024, 12(8), 1164; https://doi.org/10.3390/math12081164 - 12 Apr 2024
Cited by 1 | Viewed by 1227
Abstract
We illustrate how structural equation models (SEMs) can be used to assess the reliability and generalizability of composite and subscale scores, proportions of multiple sources of measurement error, and subscale added value within multivariate designs using data from a popular inventory measuring hierarchically [...] Read more.
We illustrate how structural equation models (SEMs) can be used to assess the reliability and generalizability of composite and subscale scores, proportions of multiple sources of measurement error, and subscale added value within multivariate designs using data from a popular inventory measuring hierarchically structured personality traits. We compare these techniques between standard SEMs representing congeneric relations between indicators and underlying factors versus SEM-based generalizability theory (GT) designs with simplified essential tau-equivalent constraints. Results strongly emphasized the importance of accounting for multiple sources of measurement error in both contexts and revealed that, in most but not all instances, congeneric designs yielded higher score accuracy, lower proportions of measurement error, greater average subscale score viability, stronger model fits, and differing magnitudes of disattenuated subscale intercorrelations. Extending the congeneric analyses to the item level further highlighted consistent weaknesses in the psychometric properties of negatively versus positively keyed items. Collectively, these findings demonstrate the practical value and advantages of applying GT-based principles to congeneric SEMs that are much more commonly encountered in the research literature and more directly linked to the specific measures being analyzed. We also provide prophecy formulas to estimate reliability and generalizability coefficients, proportions of individual sources of measurement error, and subscale added-value indices for changes made to measurement procedures and offer guidelines and examples for running all illustrated analyses using the lavaan (Version 0.6-17) and semTools (Version 0.5-6) packages in R. The methods described for the analyzed designs are applicable to any objectively or subjectively scored assessments for which both composite and subcomponent scores are reported. Full article
(This article belongs to the Special Issue Reliability Estimation and Mathematical Statistics)
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15 pages, 2474 KiB  
Article
Traffic Safety Assessment and Injury Severity Analysis for Undivided Two-Way Highway–Rail Grade Crossings
by Qiaoqiao Ren, Min Xu, Bojian Zhou and Sai-Ho Chung
Mathematics 2024, 12(4), 519; https://doi.org/10.3390/math12040519 - 7 Feb 2024
Cited by 5 | Viewed by 1718
Abstract
The safety and reliability of undivided two-way highway–rail grade crossings (HRGCs) are of paramount importance in transportation systems. Utilizing crash data from the Federal Railroad Administration between 2020 and 2021, this study aims to predict crash injury severity outcomes and investigate various factors [...] Read more.
The safety and reliability of undivided two-way highway–rail grade crossings (HRGCs) are of paramount importance in transportation systems. Utilizing crash data from the Federal Railroad Administration between 2020 and 2021, this study aims to predict crash injury severity outcomes and investigate various factors influencing injury severities. The χ2 test was first used to select variables that were significantly associated with injury outcomes. By employing the eXtreme Gradient Boosting (XGBoost) model and interpretable SHapley Additive exPlanations (SHAP), a cross-category safety assessment that offers an evidence-based hierarchy and statistical inference of risk factors associated with crashes, crossings, vehicles, drivers, and environment was provided for killed, injured, and uninjured outcomes. Some significant predictors overlapped between the killed and injured models, such as old driver, driver was in vehicle, main track, went around the gate, adverse crossing surface, and truck, while the other different significant factors revealed that the model could distinguish between different severity levels. Additionally, the results suggested that the model has varying performances in predicting different injury severities, with the killed model having the highest accuracy of 93.36%. The SHAP dependency plots for the top three features also ensure reliable predictions and inform potential interventions aimed at strengthening traffic safety and risk management practices, such as enhanced warning systems and targeted educational campaigns for older drivers. Full article
(This article belongs to the Special Issue Reliability Estimation and Mathematical Statistics)
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