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Symmetry
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Symmetrical and Asymmetrical Distributions in Statistics and Probability
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Symmetrical and Asymmetrical Distributions in Statistics and Probability

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (15 August 2023) | Viewed by 20688

Special Issue Editors

Dr. Piao Chen

E-Mail Website
Guest Editor
Department of Applied Mathematics, Delft University of Technology, 2600 AA Delft, The Netherlands
Interests: complex system modelling; maintenance planning; reliability and resilience
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Ancha Xu

E-Mail Website
Guest Editor
Department of Statistics, Zhejiang Gongshang University, Zhejiang, Hangzhou, China
Interests: degradation model; failure data analysis; Bayesian computation

Special Issue Information

Dear Colleagues,

Probability distribution is one of the most important concepts in probability and statistics that provides a mathematical description of random events. Probability distributions are ubiquitous in statistical applications, and their application areas include but are not limited to engineering, life science, medicine, environments, social science, and natural science. Generally, probability distributions can be categorised into two groups, i.e., symmetrical and asymmetrical distributions. A symmetrical distribution is a type of distribution whose probability density function or probability mass function is symmetric around its mean. The most famous example of the symmetric distributions is the normal distribution, and other examples include the t-distribution, the Cauchy distribution, the logistic distribution, the uniform distribution, etc. On the other hand, the shape of an asymmetrical distribution is not symmetric, and examples include the gamma distribution, the log-normal distribution, the beta distribution, the Weibull distribution, etc. Both symmetrical and asymmetrical distributions have received considerable attention in the probability and statistics literature.

In this Special Issue, we are interested in theoretical developments and new applications of symmetrical and asymmetrical distributions in all areas. In particular, the authors need to highlight the concept of symmetry in their studies. This can be done either by explaining why a symmetrical or asymmetrical distribution is appropriate for their applications or by investigating the differences between these two types of distribution in their studies.

Dr. Piao Chen
Prof. Dr. Ancha Xu
Guest Editors

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. Symmetry is an international peer-reviewed open access monthly 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 2400 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
  • probability theory
  • statistical application
  • reliability

Published Papers (15 papers)

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Research

13 pages, 1043 KiB  
Article
A New Two-Parameter Discrete Distribution for Overdispersed and Asymmetric Data: Its Properties, Estimation, Regression Model, and Applications
by Amani Alrumayh and Hazar A. Khogeer
Symmetry 2023, 15(6), 1289; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15061289 - 20 Jun 2023
Cited by 2 | Viewed by 1243
Abstract
A novel discrete Poisson mixing probability distribution with two parameters has been developed by combining the Poisson distribution with the transmuted moment exponential distribution. It is possible to deduce several mathematical properties, such as the moment-generating function, ordinary moments, moments about the mean, [...] Read more.
A novel discrete Poisson mixing probability distribution with two parameters has been developed by combining the Poisson distribution with the transmuted moment exponential distribution. It is possible to deduce several mathematical properties, such as the moment-generating function, ordinary moments, moments about the mean, skewness, kurtosis, and the dispersion index. The maximum likelihood estimation method is utilized to estimate the model’s parameters. A thorough simulation study is utilized to determine the behavior of the generated estimators. Estimating model parameters using a Bayesian methodology is another primary topic of this research. The behavior of Bayesian estimates is evaluated by first charting the trace, then generating 1,005,000 iterations of the Markov chain Monte Carlo method. In addition to this, we suggest a new count regression model that uses Poisson and negative binomial models in an alternating fashion. In conclusion, asymmetric datasets derived from various research areas are utilized for practical applications. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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13 pages, 320 KiB  
Article
Triple Sampling Inference Procedures for the Mean of the Normal Distribution When the Population Coefficient of Variation Is Known
by Ali Alhajraf, Ali Yousef and Hosny Hamdy
Symmetry 2023, 15(3), 672; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15030672 - 07 Mar 2023
Viewed by 929
Abstract
This paper discusses the triple sampling inference procedures for the mean of a symmetric distribution—the normal distribution when the coefficient of variation is known. We use the Searls’ estimator as an initial estimate for the unknown population mean rather than the classical sample [...] Read more.
This paper discusses the triple sampling inference procedures for the mean of a symmetric distribution—the normal distribution when the coefficient of variation is known. We use the Searls’ estimator as an initial estimate for the unknown population mean rather than the classical sample mean. In statistics literature, the normal distribution under investigation underlines almost all the natural phenomena with applications in many fields. First, we discuss the minimum risk point estimation problem under a squared error loss function with linear sampling cost. We obtained all asymptotic results that enhanced finding the second-order asymptotic risk and regret. Second, we construct a fixed-width confidence interval for the mean that satisfies at least a predetermined nominal value and find the second-order asymptotic coverage probability. Both estimation problems are performed under a unified optimal framework. The theoretical results reveal that the performance of the triple sampling procedure depends on the numerical value of the coefficient of variation—the smaller the coefficient of variation, the better the performance of the procedure. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
18 pages, 2449 KiB  
Article
Failure Evaluation of Electronic Products Based on Double Hierarchy Hesitant Fuzzy Linguistic Term Set and K-Means Clustering Algorithm
by Jinkun Dai, Jihong Pang, Qiang Luo and Qianbing Huang
Symmetry 2022, 14(12), 2555; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14122555 - 03 Dec 2022
Cited by 3 | Viewed by 1048
Abstract
The extent of failure evaluation depends on the complexity and significance of electronic products. However, traditional failure mode and effect analysis (FMEA) has many shortcomings, which brings large difficulty to failure evaluating work. This paper uses the double hierarchy hesitant fuzzy linguistic term [...] Read more.
The extent of failure evaluation depends on the complexity and significance of electronic products. However, traditional failure mode and effect analysis (FMEA) has many shortcomings, which brings large difficulty to failure evaluating work. This paper uses the double hierarchy hesitant fuzzy linguistic term set (DHHFLTS) and the K-means clustering algorithm to improve the shortcomings of traditional FMEA. The DHHFLTS is a set of language terms based on the symmetry method and uniform language granularity. Firstly, we determine the product failure mode and set up an evaluation team after formulating an evaluation symmetrical language set. The psychological changes of the evaluators can be truly expressed by using the DHHFLTS. Secondly, the entropy weight method is used to calculate the weight of the evaluation members. The evaluation information of the evaluation personnel on the failure mode is aggregated based on the weight of the evaluation members. Then, the K-means clustering algorithm is used to calculate the distance between failure modes and each cluster center point by using the normalized weight of influencing factors and the evaluation distance of each evaluator. Finally, the evaluation of an electromagnet failure mode is taken as an example to prove the objectivity and practicability of the new method. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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26 pages, 659 KiB  
Article
General Entropy with Bayes Techniques under Lindley and MCMC for Estimating the New Weibull–Pareto Parameters: Theory and Application
by Mohamed S. Eliwa, Rashad M. EL-Sagheer, Samah H. El-Essawy, Bader Almohaimeed, Fahad S. Alshammari and Mahmoud El-Morshedy
Symmetry 2022, 14(11), 2395; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14112395 - 12 Nov 2022
Cited by 2 | Viewed by 1103
Abstract
Censored data play a pivotal role in life testing experiments since they significantly reduce cost and testing time. Hence, this paper investigates the problem of statistical inference for a system of progressive first-failure censoring data for a new Weibull–Pareto distribution. Maximum likelihood estimates [...] Read more.
Censored data play a pivotal role in life testing experiments since they significantly reduce cost and testing time. Hence, this paper investigates the problem of statistical inference for a system of progressive first-failure censoring data for a new Weibull–Pareto distribution. Maximum likelihood estimates for the parameters as well as some lifetime indices such as reliability, hazard rate functions, and coefficient of variation are derived. Lindley approximation and the Markov chain Monte Carlo technique are applied to obtain the Bayes estimates relative to two different loss functions: balanced linear exponential and general entropy loss functions. The results of the Bayes estimate are computed under the consideration of informative prior function. A real-life example "the survival times in years of a group of patients given chemotherapy treatment" is presented to illustrate the proposed methods. Finally, a simulation study is carried out to determine the performance of the maximum likelihood and Bayes estimates and compare the performance of different corresponding confidence intervals. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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21 pages, 730 KiB  
Article
Reliability Inferences of the Inverted NH Parameters via Generalized Type-II Progressive Hybrid Censoring with Applications
by Ahmed Elshahhat, Heba S. Mohammed and Osama E. Abo-Kasem
Symmetry 2022, 14(11), 2379; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14112379 - 10 Nov 2022
Cited by 5 | Viewed by 1131
Abstract
Generalized progressive hybrid censored mechanisms have been proposed to reduce the test duration and to save the cost spent on testing. This paper considers the problem of estimating the unknown model parameters and the reliability time functions of the new inverted Nadarajah–Haghighi (NH) [...] Read more.
Generalized progressive hybrid censored mechanisms have been proposed to reduce the test duration and to save the cost spent on testing. This paper considers the problem of estimating the unknown model parameters and the reliability time functions of the new inverted Nadarajah–Haghighi (NH) distribution under generalized Type-II progressive hybrid censoring using the maximum likelihood and Bayesian estimation approaches. Utilizing the normal approximation of the frequentist estimators, the corresponding approximate confidence intervals of unknown quantities are also constructed. Using independent gamma conjugate priors under the symmetrical squared error loss, the Bayesian estimators are developed. Since the joint likelihood function is obtained in complex form, the Bayesian estimators and their associated highest posterior density intervals cannot be obtained analytically but can be evaluated via Monte Carlo Markov chain techniques. To select the optimum censoring scheme among different censoring plans, five optimality criteria are used. Finally, to explain how the proposed methodologies can be applied in real situations, two applications representing the failure times of electronic devices and deaths from the coronavirus disease 2019 epidemic in the United States of America are analyzed. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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14 pages, 327 KiB  
Article
Confidence Intervals for Common Coefficient of Variation of Several Birnbaum–Saunders Distributions
by Wisunee Puggard, Sa-Aat Niwitpong and Suparat Niwitpong
Symmetry 2022, 14(10), 2101; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14102101 - 09 Oct 2022
Cited by 5 | Viewed by 995
Abstract
The Birnbaum–Saunders (BS) distribution, also known as the fatigue life distribution, is right-skewed and used to model the failure times of industrial components. It has received much attention due to its attractive properties and its relationship to the normal distribution (which is symmetric). [...] Read more.
The Birnbaum–Saunders (BS) distribution, also known as the fatigue life distribution, is right-skewed and used to model the failure times of industrial components. It has received much attention due to its attractive properties and its relationship to the normal distribution (which is symmetric). Furthermore, the coefficient of variation (CV) is commonly used to analyze variation within a dataset. In some situations, the independent samples are collected from different instruments or laboratories. Consequently, it is of importance to make inference for the common CV. To this end, confidence intervals based on the generalized confidence interval (GCI), method of variance estimates recovery (MOVER), large-sample (LS), Bayesian credible interval (BayCrI), and highest posterior density interval (HPDI) methods are proposed herein to estimate the common CV of several BS distributions. Their performances in terms of their coverage probabilities and average lengths were investigated by using Monte Carlo simulation. The simulation results indicate that the HPDI-based confidence interval outperformed the others in all of the investigated scenarios. Finally, the efficacies of the proposed confidence intervals are illustrated by applying them to real datasets of PM10 (particulate matter ≤ 10 μm) concentrations from three pollution monitoring stations in Chiang Mai, Thailand. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
13 pages, 308 KiB  
Article
Robust Sparse Reduced-Rank Regression with Response Dependency
by Wenchen Liu, Guanfu Liu and Yincai Tang
Symmetry 2022, 14(8), 1617; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081617 - 06 Aug 2022
Viewed by 1029
Abstract
In multiple response regression, the reduced rank regression model is an effective method to reduce the number of model parameters and it takes advantage of interrelation among the response variables. To improve the prediction performance of the multiple response regression, a method for [...] Read more.
In multiple response regression, the reduced rank regression model is an effective method to reduce the number of model parameters and it takes advantage of interrelation among the response variables. To improve the prediction performance of the multiple response regression, a method for the sparse robust reduced rank regression with covariance estimation(Cov-SR4) is proposed, which can carry out variable selection, outlier detection, and covariance estimation simultaneously. The random error term of this model follows a multivariate normal distribution which is a symmetric distribution and the covariance matrix or precision matrix must be a symmetric matrix that reduces the number of parameters. Both the element-wise penalty function and row-wise penalty function can be used to handle different types of outliers. A numerical algorithm with a covariance estimation method is proposed to solve the robust sparse reduced rank regression. We compare our method with three recent reduced rank regression methods in a simulation study and real data analysis. Our method exhibits competitive performance both in prediction error and variable selection accuracy. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
16 pages, 482 KiB  
Article
A Continuous Granular Model for Stochastic Reserving with Individual Information
by Zhigao Wang and Wenchen Liu
Symmetry 2022, 14(8), 1582; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081582 - 01 Aug 2022
Viewed by 858
Abstract
This paper works on the claims data generated by individual policies which are randomly exposed to a period of continuous time. The main aim is to model the occurrence times of individual claims, as well as their developments given the feature information and [...] Read more.
This paper works on the claims data generated by individual policies which are randomly exposed to a period of continuous time. The main aim is to model the occurrence times of individual claims, as well as their developments given the feature information and exposure periods of individual policies, and thus project the outstanding liabilities. In this paper, we also propose a method to compute the moments of outstanding liabilities in an analytic form. It is significant for a general insurance company to more accurately project outstanding liabilities in risk management. It is well-known that the features of individual policies have effects on the occurrence of claims and their developments and thus the projection of outstanding liabilities. Neglecting the information can unquestionably decrease the prediction accuracy of stochastic reserving, where the accuracy is measured by the mean square error of prediction (MSEP), whose analytic form is computed according to the derived moments of outstanding liabilities. The parameters concerned in the proposed model are estimated based on likelihood and quasi-likelihood and the properties of estimated parameters are further studied. The asymptotic behavior of stochastic reserving is also investigated. The asymptotic distribution of parameter estimators is multivariate normal distribution which is a symmetric distribution and the asymptotic distribution of the deviation of the estimated loss reserving from theoretical loss reserve also follows a normal distribution. The confidence intervals for the parameter estimators and the deviation can be easily obtained through the symmetry of the normal distribution. Some simulations are conducted in order to support the main theoretical results. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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22 pages, 1029 KiB  
Article
Complexity Analysis of E-Bayesian Estimation under Type-II Censoring with Application to Organ Transplant Blood Data
by Mazen Nassar, Refah Alotaibi and Ahmed Elshahhat
Symmetry 2022, 14(7), 1308; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071308 - 24 Jun 2022
Cited by 4 | Viewed by 1121
Abstract
The E-Bayesian estimation approach has been presented for estimating the parameter and/or reliability characteristics of various models. Several investigations in the literature have considered this method under the assumption that just one parameter is unknown. So, based on Type-II censoring, this study proposes [...] Read more.
The E-Bayesian estimation approach has been presented for estimating the parameter and/or reliability characteristics of various models. Several investigations in the literature have considered this method under the assumption that just one parameter is unknown. So, based on Type-II censoring, this study proposes for the first time an effort to use the E-Bayesian estimation approach to estimate the full model parameters as well as certain related functions such as the reliability and hazard rate functions. To illustrate this purpose, we apply the proposed technique to the two-parameter generalized inverted exponential distribution which can be considered to be one of the most flexible asymmetrical probability distributions. Moreover, the E-Bayesian method, maximum likelihood, and Bayesian estimation approaches are also considered for comparison purposes. Under the assumption of independent gamma priors, the Bayes and E-Bayes estimators are developed using the symmetrical squared error loss function. Due to the complex form of the joint posterior density, two approximation techniques, namely the Lindley and Markov chain Monte Carlo methods, are considered to carry out the Bayes and E-Bayes estimates and also to construct the associate credible intervals. Monte Carlo simulations are performed to assess the performance of the proposed estimators. To demonstrate the applicability of the proposed methods in real phenomenon, one real data set is analyzed and it shows that the proposed method is effective and easy to operate in a real-life scenario. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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21 pages, 390 KiB  
Article
Robust Optimum Life-Testing Plans under Progressive Type-I Interval Censoring Schemes with Cost Constraint
by Xiaodong Zhou, Yunjuan Wang and Rongxian Yue
Symmetry 2022, 14(5), 1047; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14051047 - 19 May 2022
Viewed by 1213
Abstract
This paper considers optimal design problems for the Weibull distribution, which can be used to model symmetrical or asymmetrical data, in the presence of progressive interval censoring in life-testing experiments. Two robust approaches, Bayesian and minimax, are proposed to deal with the dependence [...] Read more.
This paper considers optimal design problems for the Weibull distribution, which can be used to model symmetrical or asymmetrical data, in the presence of progressive interval censoring in life-testing experiments. Two robust approaches, Bayesian and minimax, are proposed to deal with the dependence of the D-optimality and c-optimality on the unknown model parameters. Meanwhile, the compound design method is applied to ensure a compromise between the precision of estimation of the model parameters and the precision of estimation of the quantiles. Furthermore, to make the design become more practical, the cost constraints are taken into account in constructing the optimal designs. Two algorithms are provided for finding the robust optimal solutions. A simulated example and a real life example are given to illustrate the proposed methods. The sensitivity analysis is also studied. These new design methods can help the engineers to obtain robust optimal designs for the censored life-testing experiments. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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20 pages, 3258 KiB  
Article
A New Dynamic Fault Tree Analysis Method of Electromagnetic Brakes Based on Bayesian Network Accompanying Wiener Process
by Jihong Pang, Jinkun Dai, Chaohui Zhang, Hongyong Zhou and Yong Li
Symmetry 2022, 14(5), 968; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14050968 - 09 May 2022
Cited by 2 | Viewed by 1604
Abstract
Product fault diagnosis has always been the focus of quality and reliability research. However, a failure–rate curve of some products is a symmetrical function, the fault analysis result is not true because the failure period of the products cannot be judged accurately. In [...] Read more.
Product fault diagnosis has always been the focus of quality and reliability research. However, a failure–rate curve of some products is a symmetrical function, the fault analysis result is not true because the failure period of the products cannot be judged accurately. In order to solve the problem of fault diagnosis, this paper proposes a new Takagi-Sugeno (T-S) dynamic fault tree analysis method based on a Bayesian network accompanying the Wiener process. Firstly, the top event, middle event, and bottom event of the product failure mode are determined, and the T-S dynamic fault tree is constructed. Secondly, in order to form the Bayesian network diagram of the T-S dynamic fault tree, the events in the fault tree are transformed into nodes, and the T-S dynamic gate is also transformed into directed edges. Then, the Wiener process is used to model the performance degradation process of the stationary independent increment of the symmetric function distribution, and the maximum likelihood estimation method is applied to estimate the unknown parameters of the degradation model. Next, the product residual life prediction model is established based on the concept of first arrival time, and a symmetric function of failure–rate curve is obtained by using the product failure probability density function. According to the fault density function derived from the Wiener process, the reverse reasoning algorithm of the Bayesian network is established. Combined with the prior probability of the bottom event, the posterior probability of the root node is calculated and sorted as well. Finally, taking the insufficient braking force of electromagnetic brakes as an example, the practicability and objectivity of the new method are proved. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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32 pages, 7166 KiB  
Article
Interval Modeling for Gamma Process Degradation Model
by Guihong Liu, Qiang Guan, Yincai Tang and Yunhuei Tzeng
Symmetry 2022, 14(5), 954; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14050954 - 07 May 2022
Cited by 3 | Viewed by 1387
Abstract
In this paper, we proposed an interval degradation model to improve the reliability of the classical single point degradation model. The interval degradation model is very flexible when model parameters follows different distributions. Twenty-five types of interval Gamma degradation models are considered and [...] Read more.
In this paper, we proposed an interval degradation model to improve the reliability of the classical single point degradation model. The interval degradation model is very flexible when model parameters follows different distributions. Twenty-five types of interval Gamma degradation models are considered and discussed under different conditions. The reliabilities of interval Gamma degradation models are obtained. The Monte Carlo method has been studied to compute the reliability and lifetime of interval Gamma degradation model. The numerical examples are conducted to compare the interval degradation model with the classical single point degradation model. Simulation results reveal that the performance of reliability and mean lifetime of interval Gamma degradation model are much better than those of the single Gamma degradation model. Finally, we applied our model to a real data example and demonstrated the effectiveness and feasibility of the interval Gamma degradation model. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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21 pages, 2211 KiB  
Article
Inferences for Alpha Power Exponential Distribution Using Adaptive Progressively Type-II Hybrid Censored Data with Applications
by Refah Alotaibi, Ahmed Elshahhat, Hoda Rezk and Mazen Nassar
Symmetry 2022, 14(4), 651; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14040651 - 23 Mar 2022
Cited by 18 | Viewed by 1696
Abstract
One of the most important asymmetrical probability distributions that recently presented as an extension of the conventional exponential distribution is the alpha power exponential distribution. It may be compared to various asymmetrical well-known models, such as Weibull and gamma distributions. As a result, [...] Read more.
One of the most important asymmetrical probability distributions that recently presented as an extension of the conventional exponential distribution is the alpha power exponential distribution. It may be compared to various asymmetrical well-known models, such as Weibull and gamma distributions. As a result, using an adaptive progressive Type-II hybrid censoring scheme, this paper investigates the estimation problems of the alpha power exponential distribution. Maximum likelihood and Bayesian methods are used to estimate unknown parameters, reliability, and hazard rate functions. Under the assumption of independent gamma priors and symmetric loss function, Bayesian estimators are examined. The Bayesian credible intervals and estimated confidence intervals of the relevant values are also calculated. The various estimating approaches are evaluated using a simulation study that considers various sample sizes and censoring schemes. Furthermore, numerous optimality criteria are examined, and the best progressive censoring schemes are offered. Finally, for an explanation, two real data sets from engineering and chemical fields are provided to show the applicability of the asymmetrical alpha power exponential distribution. The Bayesian method for estimating the parameters and reliability indices of the alpha power exponential distribution is recommended based on numerical results, especially when the number of observed data is small. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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17 pages, 363 KiB  
Article
Mean Remaining Strength Estimation of Multi-State System Based on Nonparametric Bayesian Method
by Bin Liu, Meiling Huo, Jing Xu, Xueying Cui and Xiufeng Xie
Symmetry 2022, 14(3), 555; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14030555 - 10 Mar 2022
Cited by 1 | Viewed by 1409
Abstract
In a stress-strength system, the mean remaining strength is the key to deciding the safety threshold for the system continuing to operate. In this study, a multi-state stress-strength system composed of two-state components is discussed, and the mean remaining strength of the system [...] Read more.
In a stress-strength system, the mean remaining strength is the key to deciding the safety threshold for the system continuing to operate. In this study, a multi-state stress-strength system composed of two-state components is discussed, and the mean remaining strength of the system is studied. Applying a multidimensional signature, the dynamic signature form is established, and the mean remaining strength of the system in different states is deduced. Moreover, the nonparametric Bayesian method is used to estimate the mean remaining strength of the system. The results of Monte Carlo simulation show that the nonparametric Bayesian method can reasonably estimate the mean remaining strength of a multi-state system, and its estimation effect is better than that of the nonparametric estimation method. A practical case based on a fiber strength dataset is presented as an application of the proposed methodology. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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20 pages, 385 KiB  
Article
Inference for Kumaraswamy Distribution under Generalized Progressive Hybrid Censoring
by Liang Wang, Ying Zhou, Yuhlong Lio and Yogesh Mani Tripathi
Symmetry 2022, 14(2), 403; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020403 - 17 Feb 2022
Cited by 7 | Viewed by 1284
Abstract
In this paper, generalized progressive hybrid censoring is discussed, while a scheme is designed to provide a flexible and symmetrical scenario to collect failure information in the whole life cycle of units. When the lifetime of units follows Kumaraswamy distribution, inference is investigated [...] Read more.
In this paper, generalized progressive hybrid censoring is discussed, while a scheme is designed to provide a flexible and symmetrical scenario to collect failure information in the whole life cycle of units. When the lifetime of units follows Kumaraswamy distribution, inference is investigated under classical and Bayesian approaches. The maximum likelihood estimates and associated existence and uniqueness properties are established and the confidence intervals for unknown parameters are provided by using a large sample size based on asymptotic theory. Moreover, the Bayes estimates along with highest probability density credible intervals are also developed through the Monte-Carlo Markov Chain sampling technique to approximate the associated posteriors. Simulation studies and a real-life example are presented for illustration purposes. Full article
(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)
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