Symmetry in Probability Theory and Statistics

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

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 8643

Special Issue Editors

School of Mathematics and Statistics, Qingdao University, Qingdao 266000, China
Interests: statistical inference; model averaging; fiducial inference; generalized inference; mixed linear model; reliability analysis; data mining; statistical computing; computer experiment design
Department of Statistics, Anhui Normal University, Wuhu 241003, China
Interests: mathematical statistics; applied statistics
College of Big Data and Internet, Shenzhen Technology University, Shenzhen 518118, China
Interests: change point detection; distribution theory; statistical analysis; nonparametric estimation; skew normal distribution; time series model; network regression model; outsourced computing

Special Issue Information

Dear Colleagues,

In this Special Issue of Symmetry, we are aiming to present research and articles on the concept of symmetry in probability theory and statistics.

Symmetry plays an important role in all aspects of natural sciences, e.g., in econometrics, engineering science, computer science and biology, etc.

Symmetry is a central notion in probability theory and statistics, appearing in various forms in distribution theory, modeling, nonparametric estimation, parametric estimation, statistical tests, change point detection, model averaging, Bayesian analysis, fiducial inference, variable selection, as well as in many other branches of modern interest. The objective of this Special Issue is to publish highly motivated, original, and innovative research articles that apply the notion of symmetry to current topics in probability theory and statistics.

The scope of this issue includes, but is not limited to, the following topics:

  • Skew distributions
  • Estimation/ Nonparametric estimation
  • Change point detection;
  • Model averaging;
  • Dimension reduction and variable selection;
  • Bayesian analysis;
  • Fiducial inference;
  • Distribution theory;
  • Kriging regression model;
  • Mixed model;
  • Modelling;
  • Statistical algorithms;
  • Stochastic processes and applications;
  • Time series analysis.

Prof. Dr. Xinmin Li
Prof. Dr. Daojiang He
Dr. Weizhong Tian
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

  • elliptical symmetry
  • rotational symmetry
  • skew distributions
  • tests for symmetry
  • model averaging
  • fiducial inference
  • variable selection
  • mixed model
  • Kriging regression model

Published Papers (10 papers)

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Research

16 pages, 1501 KiB  
Article
Wald Intervals via Profile Likelihood for the Mean of the Inverse Gaussian Distribution
by Patchanok Srisuradetchai, Ausaina Niyomdecha and Wikanda Phaphan
Symmetry 2024, 16(1), 93; https://doi.org/10.3390/sym16010093 - 11 Jan 2024
Cited by 1 | Viewed by 1002
Abstract
The inverse Gaussian distribution, known for its flexible shape, is widely used across various applications. Existing confidence intervals for the mean parameter, such as profile likelihood, reparametrized profile likelihood, and Wald-type reparametrized profile likelihood with observed Fisher information intervals, are generally effective. However, [...] Read more.
The inverse Gaussian distribution, known for its flexible shape, is widely used across various applications. Existing confidence intervals for the mean parameter, such as profile likelihood, reparametrized profile likelihood, and Wald-type reparametrized profile likelihood with observed Fisher information intervals, are generally effective. However, our simulation study identifies scenarios where the coverage probability falls below the nominal confidence level. Wald-type intervals are widely used in statistics and have a symmetry property. We mathematically derive the Wald-type profile likelihood (WPL) interval and the Wald-type reparametrized profile likelihood with expected Fisher information (WRPLE) interval and compare their performance to existing methods. Our results indicate that the WRPLE interval outperforms others in terms of coverage probability, while the WPL typically yields the shortest interval. Additionally, we apply these proposed intervals to a real dataset, demonstrating their potential applicability to other datasets that follow the IG distribution. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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14 pages, 610 KiB  
Article
Different Methods for Estimating Default Parameters of Alpha Power-Transformed Power Distributions Using Record-Breaking Data
by Rasha Abd El-Wahab Attwa and Taha Radwan
Symmetry 2024, 16(1), 30; https://doi.org/10.3390/sym16010030 - 26 Dec 2023
Viewed by 649
Abstract
The current study addresses the estimation of the default parameters of alpha power-transformed power (APTPO) distributions. For the location and scale parameters of the APTPO distributions, we provide coefficients for both the best linear unbiased estimators (BLUE) and the best linear invariant estimators [...] Read more.
The current study addresses the estimation of the default parameters of alpha power-transformed power (APTPO) distributions. For the location and scale parameters of the APTPO distributions, we provide coefficients for both the best linear unbiased estimators (BLUE) and the best linear invariant estimators (BLIE) methods. Furthermore, we establish a forecast for future records. The parameters of the APTPO distribution are estimated using the maximum likelihood estimation method (MLE). The goodness-of-fit test (using Akaike information criterion (AIC)) is computed using both the inter-record time sequence and the entire sample. Also, we utilize a simulation approach to demonstrate the practicality and benefits of our perspective. Finally, we demonstrate the accuracy of these parameters and the performance of estimators through a real-life example. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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16 pages, 358 KiB  
Article
Estimation of Multiple Breaks in Panel Data Models Based on a Modified Screening and Ranking Algorithm
by Fuxiao Li, Yanting Xiao and Zhanshou Chen
Symmetry 2023, 15(10), 1890; https://doi.org/10.3390/sym15101890 - 09 Oct 2023
Viewed by 494
Abstract
Structural breaks are often encountered in empirical studies with large panels. This paper considers the estimation of multiple breaks in the mean of panel data model based on a modified screening and ranking algorithm. This algorithm satisfies symmetry and is suitable for both [...] Read more.
Structural breaks are often encountered in empirical studies with large panels. This paper considers the estimation of multiple breaks in the mean of panel data model based on a modified screening and ranking algorithm. This algorithm satisfies symmetry and is suitable for both cases where the jump size of break points is positive and negative. The break points are first initially screened based on the adaptive Fisher’s statistic, followed by further screening of the break points using the threshold criterion, and finally the final break points are screened using the information criterion. Furthermore, the consistency of the break point estimators is proved. The Monte Carlo simulation results show that the proposed method performs well even if the error terms are serially correlated or cross-sectionally correlated. Finally, two empirical examples illustrate the use of this method. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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31 pages, 1405 KiB  
Article
New Lifetime Distribution with Applications to Single Acceptance Sampling Plan and Scenarios of Increasing Hazard Rates
by Eberechukwu Q. Chinedu, Queensley C. Chukwudum, Najwan Alsadat, Okechukwu J. Obulezi, Ehab M. Almetwally and Ahlam H. Tolba
Symmetry 2023, 15(10), 1881; https://doi.org/10.3390/sym15101881 - 06 Oct 2023
Cited by 4 | Viewed by 858
Abstract
This article is an extension of the Chris-Jerry distribution (C-JD) in that a two-parameter Chris-Jerry distribution (TPCJD) is suggested and its characteristics are studied. Based on the determined domain of attraction and other major statistical properties, the proposed TPCJD seems to fit into [...] Read more.
This article is an extension of the Chris-Jerry distribution (C-JD) in that a two-parameter Chris-Jerry distribution (TPCJD) is suggested and its characteristics are studied. Based on the determined domain of attraction and other major statistical properties, the proposed TPCJD seems to fit into the Gumbel domain. Additionally, it has been confirmed that the stress strength is reliable. The tail study suggests that the TPCJD’s substantial tail makes it suited for a range of applications. The study took into account the single acceptance sampling approach using both simulation and real-life situations. The parameters of the TPCJD were estimated by some classical and Bayesian approaches. The mean squared errors (MSE), linear-exponential, and generalized entropy loss functions were deployed to obtain the Bayesian estimators aided by the Markov chain Monte Carlo (MCMC) simulation. An analysis of lifetime data on two events justified the use of the proposed distribution after comparing the results with some standard lifetime models. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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14 pages, 353 KiB  
Article
Log-Extended Exponential–Geometric Distribution: Moments and Inference Based on Generalized Order Statistics
by Areej M. AL-Zaydi
Symmetry 2023, 15(10), 1857; https://doi.org/10.3390/sym15101857 - 02 Oct 2023
Cited by 2 | Viewed by 672
Abstract
In this paper, we provide explicit expressions as well as recurrence relations for the single and product moments of generalized order statistics from the log-extended exponential–geometric distribution. These relations are utilized to discuss the special cases of generalized order statistics, and some numerical [...] Read more.
In this paper, we provide explicit expressions as well as recurrence relations for the single and product moments of generalized order statistics from the log-extended exponential–geometric distribution. These relations are utilized to discuss the special cases of generalized order statistics, and some numerical computations are carried out. Further, we use these results to obtain the best linear unbiased estimators for the location and scale parameters based on progressively Type-II right censored samples. Finally, a real data application is presented. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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14 pages, 310 KiB  
Article
Stochastic Comparisons of Largest-Order Statistics and Ranges from Marshall–Olkin Bivariate Exponential and Independent Exponential Variables
by Narayanaswamy Balakrishnan, Ghobad Saadat Kia (Barmalzan) and Mohammad Mehrpooya
Symmetry 2023, 15(9), 1796; https://doi.org/10.3390/sym15091796 - 20 Sep 2023
Viewed by 495
Abstract
Sample range and the associated functions such as survival function and mean residual life function have found many important applications in the reliability field. In this work, we establish some results that are in two different directions. In the first part, we establish [...] Read more.
Sample range and the associated functions such as survival function and mean residual life function have found many important applications in the reliability field. In this work, we establish some results that are in two different directions. In the first part, we establish some conditions for comparing the largest-order statistics (in the sense of mean residual life order) arising from bivariate Marshall–Olkin exponential distribution. Then, in the second part, we present some sufficient conditions for comparing sample ranges (in the sense of usual stochastic order and reversed hazard rate order) arising from independent exponential random variables. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
18 pages, 875 KiB  
Article
A More Flexible Extension of the Fréchet Distribution Based on the Incomplete Gamma Function and Applications
by Jaime S. Castillo, Mario A. Rojas and Jimmy Reyes
Symmetry 2023, 15(8), 1608; https://doi.org/10.3390/sym15081608 - 20 Aug 2023
Cited by 1 | Viewed by 774
Abstract
In this paper, a more flexible extension of the Fréchet distribution is introduced. The new distribution is defined by means of the stochastic representation as the quotient of two independent random variables, a Fréchet distribution and the power of a random variable, with [...] Read more.
In this paper, a more flexible extension of the Fréchet distribution is introduced. The new distribution is defined by means of the stochastic representation as the quotient of two independent random variables, a Fréchet distribution and the power of a random variable, with uniform distribution in the interval (0, 1). We will call this new extension the slash Fréchet distribution and one of its main characteristics is that its tails are heavier than the Fréchet distribution. The general density of this distribution and some basic properties are determined. Its moments, skewness coefficients, and kurtosis are calculated. In addition, the estimation of the model parameters is obtained by the method of moments and maximum likelihood. Finally, three applications with real data are performed by fitting the new model and comparing it with the Fréchet distribution. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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19 pages, 3359 KiB  
Article
Asymmetric Right-Skewed Size-Biased Bilal Distribution with Mathematical Properties, Reliability Analysis, Inference and Applications
by Amer Ibrahim Al-Omari, Rehab Alsultan and Ghadah Alomani
Symmetry 2023, 15(8), 1578; https://doi.org/10.3390/sym15081578 - 13 Aug 2023
Viewed by 649
Abstract
Asymmetric distributions, as opposed to symmetric distributions, may be more resilient to extreme values or outliers. Furthermore, when data show substantial skewness, asymmetric distributions can shed light on the underlying processes or phenomena being investigated. In this direction, the size-biased Bilal distribution (SBBD) [...] Read more.
Asymmetric distributions, as opposed to symmetric distributions, may be more resilient to extreme values or outliers. Furthermore, when data show substantial skewness, asymmetric distributions can shed light on the underlying processes or phenomena being investigated. In this direction, the size-biased Bilal distribution (SBBD) is suggested in this study as a generalization to the Bilal distribution. The length-biased and area-biased Bilal distributions are discussed in detail as two special cases. The main statistical properties of the distribution including the rth moment, coefficients of variation, skewness, kurtosis, moment generating function, incomplete moments, moments of residual life, harmonic mean, Fisher’s information, and the Rényi entropy as a measure of uncertainty are presented. Graphical representations of the cumulative distribution, probability density, odds, survival, hazard, reversed hazard rate, and cumulative hazard functions are presented for further explanation of the distribution behavior. In addition, the methods of moments and maximum likelihood estimates are taken into account for estimating the model parameters. A simulation study is carried out to see the efficiency of the maximum likelihood in terms of standard errors and bias. Real data sets of precipitation and myeloid leukemia patients are considered to show the practical significance of the suggested distributions as an alternative to some well-known distributions such as the Rama, Rani, Bilal, and exponential distributions. It is found that the size-biased Bilal distribution is right-skewed and has a superior fitting performance compared to the other distributions in this study. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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36 pages, 960 KiB  
Article
A New Asymmetric Modified Topp–Leone Distribution: Classical and Bayesian Estimations under Progressive Type-II Censored Data with Applications
by Mohammed Elgarhy, Najwan Alsadat, Amal S. Hassan, Christophe Chesneau and Alaa H. Abdel-Hamid
Symmetry 2023, 15(7), 1396; https://doi.org/10.3390/sym15071396 - 10 Jul 2023
Cited by 3 | Viewed by 1193
Abstract
In this article, a new modified asymmetric Topp–Leone distribution is created and developed from a theoretical and inferential point of view. It has the feature of extending the remarkable flexibility of a special one-shape-parameter lifetime distribution, known as the inverse Topp–Leone distribution, to [...] Read more.
In this article, a new modified asymmetric Topp–Leone distribution is created and developed from a theoretical and inferential point of view. It has the feature of extending the remarkable flexibility of a special one-shape-parameter lifetime distribution, known as the inverse Topp–Leone distribution, to the bounded interval [0, 1]. The probability density function of the proposed truncated distribution has the potential to be unimodal and right-skewed, with different levels of asymmetry. On the other hand, its hazard rate function can be increasingly shaped. Some important statistical properties are examined, including several different measures. In practice, the estimation of the model parameters under progressive type-II censoring is considered. To achieve this aim, the maximum likelihood, maximum product of spacings, and Bayesian approaches are used. The Markov chain Monte Carlo approach is employed to produce the Bayesian estimates under the squared error and linear exponential loss functions. Some simulation studies to evaluate these approaches are discussed. Two applications based on real-world datasets—one on the times of infection, and the second dataset is on trading economics credit rating—are considered. Thanks to its flexible asymmetric features, the new model is preferable to some known comparable models. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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20 pages, 411 KiB  
Article
Generalized Fiducial Inference for the Stress–Strength Reliability of Generalized Logistic Distribution
by Menghan Li, Liang Yan, Yaru Qiao, Xia Cai and Khamis K. Said
Symmetry 2023, 15(7), 1365; https://doi.org/10.3390/sym15071365 - 05 Jul 2023
Viewed by 816
Abstract
Generalized logistic distribution, as the generalized form of the symmetric logistic distribution, plays an important role in reliability analysis. This article focuses on the statistical inference for the stress–strength parameter R=P(Y<X) of the generalized logistic distribution [...] Read more.
Generalized logistic distribution, as the generalized form of the symmetric logistic distribution, plays an important role in reliability analysis. This article focuses on the statistical inference for the stress–strength parameter R=P(Y<X) of the generalized logistic distribution with the same and different scale parameters. Firstly, we use the frequentist method to construct asymptotic confidence intervals, and adopt the generalized inference method for constructing the generalized point estimators as well as the generalized confidence intervals. Then the generalized fiducial method is applied to construct the fiducial point estimators and the fiducial confidence intervals. Simulation results demonstrate that the generalized fiducial method outperforms other methods in terms of the mean square error, average length, and empirical coverage. Finally, three real datasets are used to illustrate the proposed methods. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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