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Open AccessEditorial

Introduction to the Special Issue in Symmetry Titled “Symmetry in Statistics and Data Science”

by Christophe Chesneau

Symmetry 2023, 15(5), 1103; https://doi.org/10.3390/sym15051103 - 18 May 2023

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Abstract

In order to introduce this Special Issue, some motivational facts are given [...] Full article

(This article belongs to the Special Issue Symmetry in Statistics and Data Science)

Research

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16 pages, 364 KiB

Open AccessArticle

Properties and Maximum Likelihood Estimation of the Novel Mixture of Fréchet Distribution

by Wikanda Phaphan, Ibrahim Abdullahi and Wirawan Puttamat

Symmetry 2023, 15(7), 1380; https://doi.org/10.3390/sym15071380 - 07 Jul 2023

Viewed by 1129

Abstract

In recent decades, there have been numerous endeavors to develop a novel category of survival distributions possessing enhanced flexibility through the extension of existing distributions. This article constructs and validates the statistical properties of a novel survival distribution in order to obtain an [...] Read more.

In recent decades, there have been numerous endeavors to develop a novel category of survival distributions possessing enhanced flexibility through the extension of existing distributions. This article constructs and validates the statistical properties of a novel survival distribution in order to obtain an alternative distribution that is suitable for analyzing survival data by presenting the novel mixture of the Fréchet distribution along with statistical properties such as the probability density function (PDF), cumulative distribution function (CDF),

r^{t h}

ordinary moment, skewness, kurtosis, moment-generating function, mean, variance, mode, survival function, hazard function, and asymptotic behavior, as well as constructing the estimators of the unknown parameter by employing the expectation-maximization (EM) algorithm, and simulated annealing. Additionally, the performance of the proposed estimators was compared with bias, mean squared errors (MSE), and simulated variances, and given an illustrative example of the proposed distribution to the survival data set in order to show that the proposed distribution is appropriate for the right-skewed data. This will be extremely advantageous in survival analysis. Full article

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23 pages, 799 KiB

Open AccessArticle

A Collection of Two-Dimensional Copulas Based on an Original Parametric Ratio Scheme

by Christophe Chesneau

Symmetry 2023, 15(5), 977; https://doi.org/10.3390/sym15050977 - 25 Apr 2023

Cited by 2 | Viewed by 1015

Abstract

The creation of two-dimensional copulas is crucial for the proposal of novel families of two-dimensional distributions and the analysis of original dependence structures between two quantitative variables. Such copulas can be developed in a variety of ways. In this article, we provide theoretical [...] Read more.

The creation of two-dimensional copulas is crucial for the proposal of novel families of two-dimensional distributions and the analysis of original dependence structures between two quantitative variables. Such copulas can be developed in a variety of ways. In this article, we provide theoretical contributions to this subject; we emphasize a new parametric ratio scheme to create copulas of the following form:

C (x, y) = (b + 1) x y / [b + ϕ (x, y)]

, where b is a constant and

ϕ (x, y)

is a two-dimensional function. As a notable feature, this form can operate an original trade-off between the product copula and more versatile copulas (not symmetric, with tail dependence, etc.). Instead of a global study, we examine seven concrete examples of such copulas, which have never been considered before. Most of them are extended versions of existing non-ratio copulas, such as the Celebioglu–Cuadras, Ali-Mikhail-Haq, and Gumbel–Barnett copulas. We discuss their attractive properties, including their symmetry, dominance, dependence, and correlation features. Some graphics and tables are given as complementary works. Our findings expand the horizons of new two-dimensional distributional or dependence modeling. Full article

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15 pages, 374 KiB

Open AccessArticle

A New Detection Function Model for Distance Sampling Based on the Burr XII Model

by Ayed R. A. Alanzi, Farrukh Jamal, Muhammad H. Tahir, Christophe Chesneau, Sana Kanwal and Waqas Sami

Symmetry 2023, 15(3), 620; https://doi.org/10.3390/sym15030620 - 01 Mar 2023

Cited by 2 | Viewed by 1029

Abstract

One of the most used techniques for determining animal abundance is distance sampling. The distance sampling framework depends on the idea of a detection function, and a number of options have been suggested. In this paper, we provide a new flexible parametric model [...] Read more.

One of the most used techniques for determining animal abundance is distance sampling. The distance sampling framework depends on the idea of a detection function, and a number of options have been suggested. In this paper, we provide a new flexible parametric model based on the Burr XII distribution. To be more specific, we use the survival function of the Burr XII distribution for novel purposes in this context. The proposed model is appealing because it meets all of the requirements for a reliable detection function model, such as being monotonically decreasing and having a shoulder at the origin. It also has the features of having various asymmetric properties and a heavy right tail, which are rare properties in this setting. In the first part, we provide its key characteristics, such as shapes and moments. Then, the inferential aspect of the model is investigated. The maximum likelihood estimation method is used to estimate the parameters in a data-fitting scenario. The estimates of population abundance are derived and compared with some existing parametric estimates. A simulation is run to assess how well the resulting estimates perform in comparison to other widely applied estimates from the literature. The model is then tested using two real-world data sets. Based on the famous goodness-of-fit statistics, we show that it is preferable to some of the well-established models. Full article

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15 pages, 324 KiB

Open AccessArticle

Robust Procedure for Change-Point Estimation Using Quantile Regression Model with Asymmetric Laplace Distribution

by Fengkai Yang

Symmetry 2023, 15(2), 447; https://doi.org/10.3390/sym15020447 - 08 Feb 2023

Cited by 1 | Viewed by 1304

Abstract

The usual mean change-point detecting method based on normal linear regression is not robust to heavy-tailed data with potential outlying points. We propose a robust change-point estimation procedure based on a quantile regression model with asymmetric Laplace error distribution and develop a non-iterative [...] Read more.

The usual mean change-point detecting method based on normal linear regression is not robust to heavy-tailed data with potential outlying points. We propose a robust change-point estimation procedure based on a quantile regression model with asymmetric Laplace error distribution and develop a non-iterative sampling algorithm from a Bayesian perspective. The algorithm can generate independently and identically distributed samples approximately from the posterior distribution of the position of the change-point, which can be used for statistical inferences straightforwardly. The procedure combines the robustness of quantile regression and the computational efficiency of the non-iterative sampling algorithm. A simulation study is conducted to illustrate the performance of the procedure with satisfying findings, and finally, real data is analyzed to show the usefulness of the algorithm by comparison with the usual change-point detection method based on normal regression. Full article

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26 pages, 1644 KiB

Open AccessArticle

Analysis of Adaptive Progressive Type-II Hybrid Censored Dagum Data with Applications

by Heba S. Mohammed, Mazen Nassar, Refah Alotaibi and Ahmed Elshahhat

Symmetry 2022, 14(10), 2146; https://doi.org/10.3390/sym14102146 - 14 Oct 2022

Cited by 7 | Viewed by 1261

Abstract

In life testing and reliability studies, obtaining whole data always takes a long time and lots of monetary and human resources. In this case, the experimenters prefer to gather data using censoring schemes that make a balance between the length of the test, [...] Read more.

In life testing and reliability studies, obtaining whole data always takes a long time and lots of monetary and human resources. In this case, the experimenters prefer to gather data using censoring schemes that make a balance between the length of the test, the desired sample size, and the cost. Lately, an adaptive progressive type-II hybrid censoring scheme is suggested to enhance the efficiency of the statistical inference. By utilizing this scheme, this paper seeks to investigate classical and Bayesian estimations of the Dagum distribution. The maximum likelihood and Bayesian estimation methods are considered to estimate the distribution parameters and some reliability indices. The Bayesian estimation is developed under the assumption of independent gamma priors and by employing symmetric and asymmetric loss functions. Due to the tough form of the joint posterior distribution, the Markov chain Monte Carlo technique is implemented to gather samples from the full conditional distributions and in turn obtain the Bayes estimates. The approximate confidence intervals and the highest posterior density credible intervals are also obtained. The effectiveness of the various suggested methods is compared through a simulated study. The optimal progressive censoring plans are also shown, and number of optimality criteria are explored. To demonstrate the applicability of the suggested point and interval estimators, two real data sets are also examined. The outcomes of the simulation study and data analysis demonstrated that the proposed scheme is adaptable and very helpful in ending the experiment when the experimenter’s primary concern is the number of failures. Full article

(This article belongs to the Special Issue Symmetry in Statistics and Data Science)

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20 pages, 477 KiB

Open AccessArticle

Statistical Inference of Weighted Exponential Distribution under Joint Progressive Type-II Censoring

by Yinuo Qiao and Wenhao Gui

Symmetry 2022, 14(10), 2031; https://doi.org/10.3390/sym14102031 - 28 Sep 2022

Cited by 4 | Viewed by 1075

Abstract

The weighted exponential distribution is a promising skewed distribution in the life-testing experiment. The joint progressive type-II censoring (JPC) scheme is an effective approach to reducing costs. In this paper, we consider the estimates of parameters of the weighted exponential distribution with the [...] Read more.

The weighted exponential distribution is a promising skewed distribution in the life-testing experiment. The joint progressive type-II censoring (JPC) scheme is an effective approach to reducing costs. In this paper, we consider the estimates of parameters of the weighted exponential distribution with the JPC data. Two populations, whose scale parameters are the same but the shape parameters of which are different, are chosen to be studied. We first evaluate the parameters with the maximum likelihood method. Because the maximum likelihood estimates of parameters cannot be obtained in closed form, we apply the Newton–Raphson method in this part. Bayesian estimates and the corresponding credible intervals under the squared error loss function are computed by using the Markov Chain Monte Carlo method. After that, we use the bootstrap method to calculate the associated confidence intervals of the unknown parameters. A simulation has been performed to test the feasibility of the above methods and real data analysis is also provided for illustrative purposes. Full article

(This article belongs to the Special Issue Symmetry in Statistics and Data Science)

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25 pages, 461 KiB

Open AccessArticle

Lagrangian Zero Truncated Poisson Distribution: Properties Regression Model and Applications

by Muhammed Rasheed Irshad, Christophe Chesneau, Damodaran Santhamani Shibu, Mohanan Monisha and Radhakumari Maya

Symmetry 2022, 14(9), 1775; https://doi.org/10.3390/sym14091775 - 25 Aug 2022

Cited by 4 | Viewed by 1546

Abstract

In this paper, we construct a new Lagrangian discrete distribution, named the Lagrangian zero truncated Poisson distribution (LZTPD). It can be presented as a generalization of the zero truncated Poissson distribution (ZTPD) and an alternative to the intervened Poisson distribution (IPD), which was [...] Read more.

In this paper, we construct a new Lagrangian discrete distribution, named the Lagrangian zero truncated Poisson distribution (LZTPD). It can be presented as a generalization of the zero truncated Poissson distribution (ZTPD) and an alternative to the intervened Poisson distribution (IPD), which was elaborated for modelling both over-dispersed and under-dispersed count datasets. The mathematical aspects of the LZTPD are thoroughly investigated, and its connection to other discrete distributions is crucially observed. Further, we define a finite mixture of LZTPDs and establish its identifiability condition along with some distributional aspects. Statistical work is then performed. The maximum likelihood and method of moment approaches are used to estimate the unknown parameters of the LZTPD. Simulation studies are also undertaken as an assessment of the long-term performance of the estimates. The significance of one additional parameter in the LZTPD is tested using a generalized likelihood ratio test. Moreover, we propose a new count regression model named the Lagrangian zero truncated Poisson regression model (LZTPRM) and its parameters are estimated by the maximum likelihood estimation method. Two real-world datasets are considered to demonstrate the LZTPD’s real-world applicability, and healthcare data are analyzed to demonstrate the LZTPRM’s superiority. Full article

(This article belongs to the Special Issue Symmetry in Statistics and Data Science)

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29 pages, 574 KiB

Open AccessArticle

On Odd Perks-G Class of Distributions: Properties, Regression Model, Discretization, Bayesian and Non-Bayesian Estimation, and Applications

by Ibrahim Elbatal, Naif Alotaibi, Ehab M. Almetwally, Salem A. Alyami and Mohammed Elgarhy

Symmetry 2022, 14(5), 883; https://doi.org/10.3390/sym14050883 - 26 Apr 2022

Cited by 31 | Viewed by 2106

Abstract

In this paper, we present a new univariate flexible generator of distributions, namely, the odd Perks-G class. Some special models in this class are introduced. The quantile function (QFUN), ordinary and incomplete moments (MOMs), generating function (GFUN), moments of residual and reversed residual [...] Read more.

In this paper, we present a new univariate flexible generator of distributions, namely, the odd Perks-G class. Some special models in this class are introduced. The quantile function (QFUN), ordinary and incomplete moments (MOMs), generating function (GFUN), moments of residual and reversed residual lifetimes (RLT), and four different types of entropy are all structural aspects of the proposed family that hold for any baseline model. Maximum likelihood (ML) and maximum product spacing (MPS) estimates of the model parameters are given. Bayesian estimates of the model parameters are obtained. We also present a novel log-location-scale regression model based on the odd Perks–Weibull distribution. Due to the significance of the odd Perks-G family and the survival discretization method, both are used to introduce the discrete odd Perks-G family, a novel discrete distribution class. Real-world data sets are used to emphasize the importance and applicability of the proposed models. Full article

(This article belongs to the Special Issue Symmetry in Statistics and Data Science)

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25 pages, 702 KiB

Open AccessArticle

Sec-G Class of Distributions: Properties and Applications

by Luciano Souza, Wilson Rosa de Oliveira, Cícero Carlos Ramos de Brito, Christophe Chesneau, Renan Fernandes and Tiago A. E. Ferreira

Symmetry 2022, 14(2), 299; https://doi.org/10.3390/sym14020299 - 01 Feb 2022

Cited by 12 | Viewed by 1647

Abstract

Although there are many continuous distributions in the literature, only a handful take advantage of the modeling power provided by trigonometric functions. To our knowledge, none of them are based on the so-called secant function, defined as the reciprocal of the cosine function. [...] Read more.

Although there are many continuous distributions in the literature, only a handful take advantage of the modeling power provided by trigonometric functions. To our knowledge, none of them are based on the so-called secant function, defined as the reciprocal of the cosine function. The secant function can go to large values whenever the cosine function goes to small values. The idea is to profit from this trigonometric property to modify well-known distribution tails and overall skewness features. With this in mind, in this paper, a new class of trigonometric distributions based on the secant function is introduced. It is called the Sec-G class. We discuss the main mathematical characteristics of this class, including series expansions of the corresponding cumulative distribution and probability density functions, as well as several probabilistic measures and functions. In particular, we present the moments, skewness, kurtosis, Lorenz, and Bonferroni curves, reliability coefficient, entropy measure, and order statistics. Throughout the study, emphasis is placed on the unique four-parameter continuous distribution of this class, defined with the Kumaraswamy-Weibull distribution as the baseline. The estimation of the model parameters is performed using the maximum likelihood method. We also carried out a numerical simulation study and present the results in graphic form. Three referenced datasets were analyzed, and it is proved that the proposed secant Kumaraswamy-Weibull model outperforms important competitors, including the Kumaraswamy-Weibull, Kumaraswamy-Weibull geometric, Kumaraswamy-Weibull Poisson, Kumaraswamy Burr XII, and Weibull models. Full article

(This article belongs to the Special Issue Symmetry in Statistics and Data Science)

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20 pages, 12543 KiB

Open AccessArticle

Spatial-Temporal 3D Residual Correlation Network for Urban Traffic Status Prediction

by Yin-Xin Bao, Quan Shi, Qin-Qin Shen and Yang Cao

Symmetry 2022, 14(1), 33; https://doi.org/10.3390/sym14010033 - 28 Dec 2021

Cited by 6 | Viewed by 1654

Abstract

Accurate traffic status prediction is of great importance to improve the security and reliability of the intelligent transportation system. However, urban traffic status prediction is a very challenging task due to the tight symmetry among the Human–Vehicle–Environment (HVE). The recently proposed spatial–temporal 3D [...] Read more.

Accurate traffic status prediction is of great importance to improve the security and reliability of the intelligent transportation system. However, urban traffic status prediction is a very challenging task due to the tight symmetry among the Human–Vehicle–Environment (HVE). The recently proposed spatial–temporal 3D convolutional neural network (ST-3DNet) effectively extracts both spatial and temporal characteristics in HVE, but ignores the essential long-term temporal characteristics and the symmetry of historical data. Therefore, a novel spatial–temporal 3D residual correlation network (ST-3DRCN) is proposed for urban traffic status prediction in this paper. The ST-3DRCN firstly introduces the Pearson correlation coefficient method to extract a high correlation between traffic data. Then, a dynamic spatial feature extraction component is constructed by using 3D convolution combined with residual units to capture dynamic spatial features. After that, based on the idea of long short-term memory (LSTM), a novel architectural unit is proposed to extract dynamic temporal features. Finally, the spatial and temporal features are fused to obtain the final prediction results. Experiments have been performed using two datasets from Chengdu, China (TaxiCD) and California, USA (PEMS-BAY). Taking the root mean square error (RMSE) as the evaluation index, the prediction accuracy of ST-3DRCN on TaxiCD dataset is 21.4%, 21.3%, 11.7%, 10.8%, 4.7%, 3.6% and 2.3% higher than LSTM, convolutional neural network (CNN), 3D-CNN, spatial–temporal residual network (ST-ResNet), spatial–temporal graph convolutional network (ST-GCN), dynamic global-local spatial–temporal network (DGLSTNet), and ST-3DNet, respectively. Full article

(This article belongs to the Special Issue Symmetry in Statistics and Data Science)

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13 pages, 1209 KiB

Open AccessArticle

Estimating the Variance of Estimator of the Latent Factor Linear Mixed Model Using Supplemented Expectation-Maximization Algorithm

by Yenni Angraini, Khairil Anwar Notodiputro, Henk Folmer, Asep Saefuddin and Toni Toharudin

Symmetry 2021, 13(7), 1286; https://doi.org/10.3390/sym13071286 - 17 Jul 2021

Cited by 1 | Viewed by 1473

Abstract

This paper deals with symmetrical data that can be modelled based on Gaussian distribution, such as linear mixed models for longitudinal data. The latent factor linear mixed model (LFLMM) is a method generally used for analysing changes in high-dimensional longitudinal data. It is [...] Read more.

This paper deals with symmetrical data that can be modelled based on Gaussian distribution, such as linear mixed models for longitudinal data. The latent factor linear mixed model (LFLMM) is a method generally used for analysing changes in high-dimensional longitudinal data. It is usual that the model estimates are based on the expectation-maximization (EM) algorithm, but unfortunately, the algorithm does not produce the standard errors of the regression coefficients, which then hampers testing procedures. To fill in the gap, the Supplemented EM (SEM) algorithm for the case of fixed variables is proposed in this paper. The computational aspects of the SEM algorithm have been investigated by means of simulation. We also calculate the variance matrix of beta using the second moment as a benchmark to compare with the asymptotic variance matrix of beta of SEM. Both the second moment and SEM produce symmetrical results, the variance estimates of beta are getting smaller when number of subjects in the simulation increases. In addition, the practical usefulness of this work was illustrated using real data on political attitudes and behaviour in Flanders-Belgium. Full article

(This article belongs to the Special Issue Symmetry in Statistics and Data Science)

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20 pages, 1139 KiB

Open AccessArticle

Bayesian Reference Analysis for the Generalized Normal Linear Regression Model

by Vera Lucia Damasceno Tomazella, Sandra Rêgo Jesus, Amanda Buosi Gazon, Francisco Louzada, Saralees Nadarajah, Diego Carvalho Nascimento, Francisco Aparecido Rodrigues and Pedro Luiz Ramos

Symmetry 2021, 13(5), 856; https://doi.org/10.3390/sym13050856 - 12 May 2021

Cited by 3 | Viewed by 2247

Abstract

This article proposes the use of the Bayesian reference analysis to estimate the parameters of the generalized normal linear regression model. It is shown that the reference prior led to a proper posterior distribution, while the Jeffreys prior returned an improper one. The [...] Read more.

This article proposes the use of the Bayesian reference analysis to estimate the parameters of the generalized normal linear regression model. It is shown that the reference prior led to a proper posterior distribution, while the Jeffreys prior returned an improper one. The inferential purposes were obtained via Markov Chain Monte Carlo (MCMC). Furthermore, diagnostic techniques based on the Kullback–Leibler divergence were used. The proposed method was illustrated using artificial data and real data on the height and diameter of Eucalyptus clones from Brazil. Full article

(This article belongs to the Special Issue Symmetry in Statistics and Data Science)

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17 pages, 389 KiB

Open AccessArticle

Computing Expectiles Using k-Nearest Neighbours Approach

by Muhammad Farooq, Sehrish Sarfraz, Christophe Chesneau, Mahmood Ul Hassan, Muhammad Ali Raza, Rehan Ahmad Khan Sherwani and Farrukh Jamal

Symmetry 2021, 13(4), 645; https://doi.org/10.3390/sym13040645 - 11 Apr 2021

Cited by 4 | Viewed by 1837

Abstract

Expectiles have gained considerable attention in recent years due to wide applications in many areas. In this study, the k-nearest neighbours approach, together with the asymmetric least squares loss function, called ex-

k N N

, is proposed for computing expectiles. Firstly, [...] Read more.

Expectiles have gained considerable attention in recent years due to wide applications in many areas. In this study, the k-nearest neighbours approach, together with the asymmetric least squares loss function, called ex-

k N N

, is proposed for computing expectiles. Firstly, the effect of various distance measures on ex-

k N N

in terms of test error and computational time is evaluated. It is found that Canberra, Lorentzian, and Soergel distance measures lead to minimum test error, whereas Euclidean, Canberra, and Average of

(L_{1}, L_{\infty})

lead to a low computational cost. Secondly, the performance of ex-

k N N

is compared with existing packages er-boost and ex-svm for computing expectiles that are based on nine real life examples. Depending on the nature of data, the ex-

k N N

showed two to 10 times better performance than er-boost and comparable performance with ex-svm regarding test error. Computationally, the ex-

k N N

is found two to five times faster than ex-svm and much faster than er-boost, particularly, in the case of high dimensional data. Full article

(This article belongs to the Special Issue Symmetry in Statistics and Data Science)

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