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Special Issue "Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II"

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

Deadline for manuscript submissions: closed (31 May 2021) | Viewed by 26023

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Special Issue Editors

Prof. Dr. Emilio Gómez Déniz

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Department of Quantitative Methods and TIDES Institute, University of Las Palmas de Gran Canaria, Campus de Tafira s/n, 35017 Las Palmas, Spain
Interests: distributions theory; bayesian statistics; robustness; bayesian applications in economics (actuarial, credibility, ruin theory)
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Dr. Enrique Calderín-Ojeda

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Guest Editor

Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Melbourne, Australia
Interests: actuarial statistics; bayesian statistics; distribution theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue is the continuation of the previous one recently published in Symmetry with the same title. Among the wide range of probability distributions available in different scenarios, the property of symmetry (and nonsymmetry) represents an important characteristic when modeling and making predictions. Obviously, current computational advances facilitate calculations that include numerous special functions that were prohibited in practical applications in the past. Hence, the catalog of probabilistic families in the literature has considerably increased.

In this Special Issue on symmetric and asymmetric distributions, researchers are invited to contribute original works and case studies related to this topic. Theoretical and applied proposals that extend the Azzalini, Jones or Marshall and Olkin schemes, or simply valid alternatives will be welcome. Authors are also encouraged to submit applied works in the field of economics (ination forecast, income and wealth, stochastic frontier models, insurance, duration models, econophysics, etc.), environmental sciences (catastrophic events, climate changes, for example), biometrics, engineering (reliability, satellite image classification, etc.), and medicine (studies related to cancer disease, cure rate models, etc.), among other areas of applications. In particular, although not limited to this, this Special Issue is intended to offer alternative methodologies to the existing modeling techniques, and it is open to original research and review articles, both theoretical and applied (empirical data adjustment, regression, Bayesian study, etc.), within the area of symmetric and asymmetric, discrete and continuous, and univariate and multivariate distributions.

Prof. Dr. Emilio Gómez Déniz
Dr. Enrique Calderín-Ojeda
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 2000 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

applications
Bayesian
kurtosis
order statistics
regression
simulation
skewness

Published Papers (16 papers)

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Research

Open AccessArticle

The Uniform Poisson–Ailamujia Distribution: Actuarial Measures and Applications in Biological Science

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Symmetry 2021, 13(7), 1258; https://doi.org/10.3390/sym13071258 - 13 Jul 2021

Cited by 9 | Viewed by 1345

Abstract

We propose a new asymmetric discrete model by combining the uniform and Poisson–Ailamujia distributions using the binomial decay transformation method. The distribution, named the uniform Poisson–Ailamujia, due to its flexibility is a good alternative to the well-known Poisson and geometric distributions for real data applications in public health, biology, sociology, medicine, and agriculture. Its main statistical properties are studied, including the cumulative and hazard rate functions, moments, and entropy. The new distribution is considered to be suitable for modeling purposes; its parameter is estimated by eight classical methods. Three applications to biological data are presented herein. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

A Generalized Rayleigh Family of Distributions Based on the Modified Slash Model

Inmaculada Barranco-Chamorro

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Symmetry 2021, 13(7), 1226; https://doi.org/10.3390/sym13071226 - 08 Jul 2021

Cited by 4 | Viewed by 1152

Abstract

Specifying a proper statistical model to represent asymmetric lifetime data with high kurtosis is an open problem. In this paper, the three-parameter, modified, slashed, generalized Rayleigh family of distributions is proposed. Its structural properties are studied: stochastic representation, probability density function, hazard rate function, moments and estimation of parameters via maximum likelihood methods. As merits of our proposal, we highlight as particular cases a plethora of lifetime models, such as Rayleigh, Maxwell, half-normal and chi-square, among others, which are able to accommodate heavy tails. A simulation study and applications to real data sets are included to illustrate the use of our results. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

Optimal Sample Size for the Birnbaum–Saunders Distribution under Decision Theory with Symmetric and Asymmetric Loss Functions

Eliardo Costa

Manoel Santos-Neto

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Víctor Leiva

Symmetry 2021, 13(6), 926; https://doi.org/10.3390/sym13060926 - 23 May 2021

Cited by 6 | Viewed by 1523

Abstract

The fatigue-life or Birnbaum–Saunders distribution is an asymmetrical model that has been widely applied in several areas of science and mainly in reliability. Although diverse methodologies related to this distribution have been proposed, the problem of determining the optimal sample size when estimating its mean has not yet been studied. In this paper, we derive a methodology to determine the optimal sample size under a decision-theoretic approach. In this approach, we consider symmetric and asymmetric loss functions for point and interval inference. Computational tools in the R language were implemented to use this methodology in practice. An illustrative example with real data is also provided to show potential applications. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications

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Symmetry 2021, 13(5), 908; https://doi.org/10.3390/sym13050908 - 20 May 2021

Cited by 2 | Viewed by 1102

Abstract

We introduce a new class of distributions called the epsilon–positive family, which can be viewed as generalization of the distributions with positive support. The construction of the epsilon–positive family is motivated by the ideas behind the generation of skew distributions using symmetric kernels. This new class of distributions has as special cases the exponential, Weibull, log–normal, log–logistic and gamma distributions, and it provides an alternative for analyzing reliability and survival data. An interesting feature of the epsilon–positive family is that it can viewed as a finite scale mixture of positive distributions, facilitating the derivation and implementation of EM–type algorithms to obtain maximum likelihood estimates (MLE) with (un)censored data. We illustrate the flexibility of this family to analyze censored and uncensored data using two real examples. One of them was previously discussed in the literature; the second one consists of a new application to model recidivism data of a group of inmates released from the Chilean prisons during 2007. The results show that this new family of distributions has a better performance fitting the data than some common alternatives such as the exponential distribution. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

A Note on the Birnbaum–Saunders Conditionals Model

Barry C. Arnold

Diego I. Gallardo

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Héctor W. Gómez

Symmetry 2021, 13(5), 762; https://doi.org/10.3390/sym13050762 - 28 Apr 2021

Cited by 1 | Viewed by 888

Abstract

As an alternative to available bivariate Birnbaum–Saunders (BS) models, a conditionally specified distribution with BS conditionals is considered. The behavior of conditional or pseudo-likelihood parameter estimates of the model parameters is investigated via simulation. A comparison using a mineralogy data set suggests that the conditionally specified model outperforms competing models (with BS marginals). An analogous comparison using a well-known data set of Australian athletes also suggests the superiority of the conditionally specified model. Further investigation of its possible general superiority is suggested. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

A New Quantile Regression for Modeling Bounded Data under a Unit Birnbaum–Saunders Distribution with Applications in Medicine and Politics

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Symmetry 2021, 13(4), 682; https://doi.org/10.3390/sym13040682 - 14 Apr 2021

Cited by 18 | Viewed by 2323

Abstract

Quantile regression provides a framework for modeling the relationship between a response variable and covariates using the quantile function. This work proposes a regression model for continuous variables bounded to the unit interval based on the unit Birnbaum–Saunders distribution as an alternative to the existing quantile regression models. By parameterizing the unit Birnbaum–Saunders distribution in terms of its quantile function allows us to model the effect of covariates across the entire response distribution, rather than only at the mean. Our proposal, especially useful for modeling quantiles using covariates, in general outperforms the other competing models available in the literature. These findings are supported by Monte Carlo simulations and applications using two real data sets. An R package, including parameter estimation, model checking as well as density, cumulative distribution, quantile and random number generating functions of the unit Birnbaum–Saunders distribution was developed and can be readily used to assess the suitability of our proposal. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

Half Logistic Inverse Lomax Distribution with Applications

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Symmetry 2021, 13(2), 309; https://doi.org/10.3390/sym13020309 - 12 Feb 2021

Cited by 1 | Viewed by 925

Abstract

The last years have revealed the importance of the inverse Lomax distribution in the understanding of lifetime heavy-tailed phenomena. However, the inverse Lomax modeling capabilities have certain limits that researchers aim to overcome. These limits include a certain stiffness in the modulation of the peak and tail properties of the related probability density function. In this paper, a solution is given by using the functionalities of the half logistic family. We introduce a new three-parameter extended inverse Lomax distribution called the half logistic inverse Lomax distribution. We highlight its superiority over the inverse Lomax distribution through various theoretical and practical approaches. The derived properties include the stochastic orders, quantiles, moments, incomplete moments, entropy (Rényi and q) and order statistics. Then, an emphasis is put on the corresponding parametric model. The parameters estimation is performed by six well-established methods. Numerical results are presented to compare the performance of the obtained estimates. Also, a simulation study on the estimation of the Rényi entropy is proposed. Finally, we consider three practical data sets, one containing environmental data, another dealing with engineering data and the last containing insurance data, to show how the practitioner can take advantage of the new half logistic inverse Lomax model. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

Extended Exponential Regression Model: Diagnostics and Application to Mineral Data

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Symmetry 2020, 12(12), 2042; https://doi.org/10.3390/sym12122042 - 10 Dec 2020

Cited by 3 | Viewed by 1254

Abstract

In this paper, we reparameterized the extended exponential model based on the mean in order to include covariates and facilitate the interpretation of the coefficients. The model is compared with common models defined in the positive line also reparametrized in the mean. Parameter estimation is approached based on the expectation–maximization algorithm. Furthermore, we discuss residuals and influence diagnostic tools. A simulation study for recovered parameters is presented. Finally, an application illustrating the advantages of the model in a real data set is presented. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

A Parametric Quantile Regression Model for Asymmetric Response Variables on the Real Line

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Symmetry 2020, 12(12), 1938; https://doi.org/10.3390/sym12121938 - 25 Nov 2020

Cited by 4 | Viewed by 1429

Abstract

In this paper, we introduce a novel parametric quantile regression model for asymmetric response variables, where the response variable follows a power skew-normal distribution. By considering a new convenient parametrization, these distribution results are very useful for modeling different quantiles of a response variable on the real line. The maximum likelihood method is employed to estimate the model parameters. Besides, we present a local influence study under different perturbation settings. Some numerical results of the estimators in finite samples are illustrated. In order to illustrate the potential for practice of our model, we apply it to a real dataset. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

Modeling of Extreme Values via Exponential Normalization Compared with Linear and Power Normalization

Haroon Mohamed Barakat

Osama Mohareb Khaled

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Nourhan Khalil Rakha

Symmetry 2020, 12(11), 1876; https://doi.org/10.3390/sym12111876 - 14 Nov 2020

Cited by 2 | Viewed by 1143

Abstract

Several new asymmetric distributions have arisen naturally in the modeling extreme values are uncovered and elucidated. The present paper deals with the extreme value theorem (EVT) under exponential normalization. An estimate of the shape parameter of the asymmetric generalized value distributions that related to this new extension of the EVT is obtained. Moreover, we develop the mathematical modeling of the extreme values by using this new extension of the EVT. We analyze the extreme values by modeling the occurrence of the exceedances over high thresholds. The natural distributions of such exceedances, new four generalized Pareto families of asymmetric distributions under exponential normalization (GPDEs), are described and their properties revealed. There is an evident symmetry between the new obtained GPDEs and those generalized Pareto distributions arisen from EVT under linear and power normalization. Estimates for the extreme value index of the four GPDEs are obtained. In addition, simulation studies are conducted in order to illustrate and validate the theoretical results. Finally, a comparison study between the different extreme models is done throughout real data sets. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess

Guillermo Martínez-Flórez

Víctor Leiva

Emilio Gómez-Déniz

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Carolina Marchant

Symmetry 2020, 12(9), 1439; https://doi.org/10.3390/sym12091439 - 01 Sep 2020

Cited by 6 | Viewed by 1710

Abstract

In this paper, we consider skew-normal distributions for constructing new a distribution which allows us to model proportions and rates with zero/one inflation as an alternative to the inflated beta distributions. The new distribution is a mixture between a Bernoulli distribution for explaining the zero/one excess and a censored skew-normal distribution for the continuous variable. The maximum likelihood method is used for parameter estimation. Observed and expected Fisher information matrices are derived to conduct likelihood-based inference in this new type skew-normal distribution. Given the flexibility of the new distributions, we are able to show, in real data scenarios, the good performance of our proposal. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

Bivariate Power-Skew-Elliptical Distribution

Guillermo Martínez-Flórez

Roger Tovar-Falón

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Héctor W. Gómez

Symmetry 2020, 12(8), 1327; https://doi.org/10.3390/sym12081327 - 09 Aug 2020

Cited by 1 | Viewed by 1218

Abstract

In this article, we introduce a power-skew-elliptical (PSE) distribution in the bivariate setting. The new bivariate model arises in the context of conditionally specified distributions. The proposed bivariate model is an absolutely continuous distribution whose marginals are univariate PSE distributions. The special case of the bivariate power-skew-normal (BPSN) distribution is studied in details. General properties of the BPSN distribution are derived and the estimation of the unknown parameters by maximum pseudo-likelihood is discussed. Further, a sandwich type matrix, which is a consistent estimator for the asymptotic covariance matrix of the maximum likelihood (ML) estimator is determined. Two applications for real data of the proposed bivariate distribution is provided for illustrative purposes. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

Approximating the Distribution of the Product of Two Normally Distributed Random Variables

Antonio Seijas-Macías

Amílcar Oliveira

Teresa A. Oliveira

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Víctor Leiva

Symmetry 2020, 12(8), 1201; https://doi.org/10.3390/sym12081201 - 22 Jul 2020

Cited by 2 | Viewed by 1799

Abstract

The distribution of the product of two normally distributed random variables has been an open problem from the early years in the XXth century. First approaches tried to determinate the mathematical and statistical properties of the distribution of such a product using different types of functions. Recently, an improvement in computational techniques has performed new approaches for calculating related integrals by using numerical integration. Another approach is to adopt any other distribution to approximate the probability density function of this product. The skew-normal distribution is a generalization of the normal distribution which considers skewness making it flexible. In this work, we approximate the distribution of the product of two normally distributed random variables using a type of skew-normal distribution. The influence of the parameters of the two normal distributions on the approximation is explored. When one of the normally distributed variables has an inverse coefficient of variation greater than one, our approximation performs better than when both normally distributed variables have inverse coefficients of variation less than one. A graphical analysis visually shows the superiority of our approach in relation to other approaches proposed in the literature on the topic. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

A Gamma-Type Distribution with Applications

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Symmetry 2020, 12(5), 870; https://doi.org/10.3390/sym12050870 - 25 May 2020

Cited by 7 | Viewed by 3354

Abstract

This article introduces a new probability distribution capable of modeling positive data that present different levels of asymmetry and high levels of kurtosis. A slashed quasi-gamma random variable is defined as the quotient of independent random variables, a generalized gamma is the numerator, and a power of a standard uniform variable is the denominator. The result is a new three-parameter distribution (scale, shape, and kurtosis) that does not present the identifiability problem presented by the generalized gamma distribution. Maximum likelihood (ML) estimation is implemented for parameter estimation. The results of two real data applications revealed a good performance in real settings. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessArticle

Bias Reduction for the Marshall-Olkin Extended Family of Distributions with Application to an Airplane’s Air Conditioning System and Precipitation Data

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Symmetry 2020, 12(5), 851; https://doi.org/10.3390/sym12050851 - 22 May 2020

Cited by 3 | Viewed by 1465

Abstract

The Marshall-Olkin extended family of distributions is an alternative for modeling lifetimes, and considers more or less asymmetry than its parent model, achieved by incorporating just one extra parameter. We investigate the bias of maximum likelihood estimators and use it to develop an estimator with less bias than traditional estimators, by a modification of the score function. Unlike other proposals, in this paper, we consider a bias reduction methodology that can be applied to any member of the family and not necessarily to any particular distribution. We conduct a Monte Carlo simulation in order to study the performance of the corrected estimators in finite samples. This simulation shows that the maximum likelihood estimator is quite biased and the proposed estimator is much less biased; in small sample sizes, the bias is reduced by around 50 percent. Two applications, related to the air conditioning system of an airplane and precipitations, are presented to illustrate the results. In those applications, the bias reduction for the shape parameters is close to 25% and the bias reduction also reduces, among others things, the width of the 95% confidence intervals for quantiles lower than 0.594. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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Open AccessFeature PaperArticle

Generalising Exponential Distributions Using an Extended Marshall–Olkin Procedure

Victoriano García

María Martel-Escobar

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F.J. Vázquez-Polo

Symmetry 2020, 12(3), 464; https://doi.org/10.3390/sym12030464 - 15 Mar 2020

Cited by 3 | Viewed by 2115

Abstract

This paper presents a three-parameter family of distributions which includes the common exponential and the Marshall–Olkin exponential as special cases. This distribution exhibits a monotone failure rate function, which makes it appealing for practitioners interested in reliability, and means it can be included in the catalogue of appropriate non-symmetric distributions to model these issues, such as the gamma and Weibull three-parameter families. Given the lack of symmetry of this kind of distribution, various statistical and reliability properties of this model are examined. Numerical examples based on real data reflect the suitable behaviour of this distribution for modelling purposes. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)

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