Research

15 pages, 1117 KiB

Open AccessArticle

by Emilio Gómez-Déniz, Yuri A. Iriarte, Enrique Calderín-Ojeda and Héctor W. Gómez

Symmetry 2019, 11(11), 1410; https://doi.org/10.3390/sym11111410 - 15 Nov 2019

Cited by 4 | Viewed by 2721

In this paper, a general class of modified power-symmetric distributions is introduced. By choosing as symmetric model the normal distribution, the modified power-normal distribution is obtained. For the latter model, some of its more relevant statistical properties are examined. Parameters estimation is carried [...] Read more.

In this paper, a general class of modified power-symmetric distributions is introduced. By choosing as symmetric model the normal distribution, the modified power-normal distribution is obtained. For the latter model, some of its more relevant statistical properties are examined. Parameters estimation is carried out by using the method of moments and maximum likelihood estimation. A simulation analysis is accomplished to study the performance of the maximum likelihood estimators. Finally, we compare the efficiency of the modified power-normal distribution with other existing distributions in the literature by using a real dataset. Full article

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

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17 pages, 799 KiB

Open AccessArticle

Normal-G Class of Probability Distributions: Properties and Applications

by Fábio V. J. Silveira, Frank Gomes-Silva, Cícero C. R. Brito, Moacyr Cunha-Filho, Felipe R. S. Gusmão and Sílvio F. A. Xavier-Júnior

Symmetry 2019, 11(11), 1407; https://doi.org/10.3390/sym11111407 - 15 Nov 2019

Cited by 5 | Viewed by 2319

Abstract

In this paper, we propose a novel class of probability distributions called Normal-G. It has the advantage of demanding no additional parameters besides those of the parent distribution, thereby providing parsimonious models. Furthermore, the class enjoys the property of identifiability whenever [...] Read more.

In this paper, we propose a novel class of probability distributions called Normal-G. It has the advantage of demanding no additional parameters besides those of the parent distribution, thereby providing parsimonious models. Furthermore, the class enjoys the property of identifiability whenever the baseline is identifiable. We present special Normal-G sub-models, which can fit asymmetrical data with either positive or negative skew. Other important mathematical properties are described, such as the series expansion of the probability density function (pdf), which is used to derive expressions for the moments and the moment generating function (mgf). We bring Monte Carlo simulation studies to investigate the behavior of the maximum likelihood estimates (MLEs) of two distributions generated by the class and we also present applications to real datasets to illustrate its usefulness. Full article

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

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17 pages, 1067 KiB

Open AccessArticle

Generalized Truncation Positive Normal Distribution

by Héctor J. Gómez, Diego I. Gallardo and Osvaldo Venegas

Symmetry 2019, 11(11), 1361; https://doi.org/10.3390/sym11111361 - 03 Nov 2019

Cited by 1 | Viewed by 2067

Abstract

In this article we study the properties, inference, and statistical applications to a parametric generalization of the truncation positive normal distribution, introducing a new parameter so as to increase the flexibility of the new model. For certain combinations of parameters, the model includes [...] Read more.

In this article we study the properties, inference, and statistical applications to a parametric generalization of the truncation positive normal distribution, introducing a new parameter so as to increase the flexibility of the new model. For certain combinations of parameters, the model includes both symmetric and asymmetric shapes. We study the model’s basic properties, maximum likelihood estimators and Fisher information matrix. Finally, we apply it to two real data sets to show the model’s good performance compared to other models with positive support: the first, related to the height of the drum of the roller and the second, related to daily cholesterol consumption. Full article

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

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10 pages, 850 KiB

Open AccessArticle

Skewness of Maximum Likelihood Estimators in the Weibull Censored Data

by Tiago M. Magalhães, Diego I. Gallardo and Héctor W. Gómez

Symmetry 2019, 11(11), 1351; https://doi.org/10.3390/sym11111351 - 01 Nov 2019

Cited by 5 | Viewed by 2108

Abstract

In this paper, we obtain a matrix formula of order

n^{- 1 / 2},

where n is the sample size, for the skewness coefficient of the distribution of the maximum likelihood estimators in the Weibull censored data. The present result is [...] Read more.

In this paper, we obtain a matrix formula of order

n^{- 1 / 2},

where n is the sample size, for the skewness coefficient of the distribution of the maximum likelihood estimators in the Weibull censored data. The present result is a nice approach to verify if the assumption of the normality of the regression parameter distribution is satisfied. Also, the expression derived is simple, as one only has to define a few matrices. We conduct an extensive Monte Carlo study to illustrate the behavior of the skewness coefficient and we apply it in two real datasets. Full article

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

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18 pages, 1048 KiB

Open AccessArticle

Flexible Birnbaum–Saunders Distribution

by Guillermo Martínez-Flórez, Inmaculada Barranco-Chamorro, Heleno Bolfarine and Héctor W. Gómez

Symmetry 2019, 11(10), 1305; https://doi.org/10.3390/sym11101305 - 16 Oct 2019

Cited by 7 | Viewed by 1991

Abstract

In this paper, we propose a bimodal extension of the Birnbaum–Saunders model by including an extra parameter. This new model is termed flexible Birnbaum–Saunders (FBS) and includes the ordinary Birnbaum–Saunders (BS) and the skew Birnbaum–Saunders (SBS) model as special cases. Its properties are [...] Read more.

In this paper, we propose a bimodal extension of the Birnbaum–Saunders model by including an extra parameter. This new model is termed flexible Birnbaum–Saunders (FBS) and includes the ordinary Birnbaum–Saunders (BS) and the skew Birnbaum–Saunders (SBS) model as special cases. Its properties are studied. Parameter estimation is considered via an iterative maximum likelihood approach. Two real applications, of interest in environmental sciences, are included, which reveal that our proposal can perform better than other competing models. Full article

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

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13 pages, 362 KiB

Open AccessArticle

An Asymmetric Distribution with Heavy Tails and Its Expectation–Maximization (EM) Algorithm Implementation

by Neveka M. Olmos, Osvaldo Venegas, Yolanda M. Gómez and Yuri A. Iriarte

Symmetry 2019, 11(9), 1150; https://doi.org/10.3390/sym11091150 - 10 Sep 2019

Cited by 3 | Viewed by 2317

Abstract

In this paper we introduce a new distribution constructed on the basis of the quotient of two independent random variables whose distributions are the half-normal distribution and a power of the exponential distribution with parameter 2 respectively. The result is a distribution with [...] Read more.

In this paper we introduce a new distribution constructed on the basis of the quotient of two independent random variables whose distributions are the half-normal distribution and a power of the exponential distribution with parameter 2 respectively. The result is a distribution with greater kurtosis than the well known half-normal and slashed half-normal distributions. We studied the general density function of this distribution, with some of its properties, moments, and its coefficients of asymmetry and kurtosis. We developed the expectation–maximization algorithm and present a simulation study. We calculated the moment and maximum likelihood estimators and present three illustrations in real data sets to show the flexibility of the new model. Full article

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

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11 pages, 2964 KiB

Open AccessArticle

An Asymmetric Bimodal Distribution with Application to Quantile Regression

by Yolanda M. Gómez, Emilio Gómez-Déniz, Osvaldo Venegas, Diego I. Gallardo and Héctor W. Gómez

Symmetry 2019, 11(7), 899; https://doi.org/10.3390/sym11070899 - 10 Jul 2019

Cited by 7 | Viewed by 3877

Abstract

In this article, we study an extension of the sinh Cauchy model in order to obtain asymmetric bimodality. The behavior of the distribution may be either unimodal or bimodal. We calculate its cumulative distribution function and use it to carry out quantile regression. [...] Read more.

In this article, we study an extension of the sinh Cauchy model in order to obtain asymmetric bimodality. The behavior of the distribution may be either unimodal or bimodal. We calculate its cumulative distribution function and use it to carry out quantile regression. We calculate the maximum likelihood estimators and carry out a simulation study. Two applications are analyzed based on real data to illustrate the flexibility of the distribution for modeling unimodal and bimodal data. Full article

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

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14 pages, 745 KiB

Open AccessArticle

Univariate and Bivariate Models Related to the Generalized Epsilon–Skew–Cauchy Distribution

by Barry C. Arnold, Héctor W. Gómez, Héctor Varela and Ignacio Vidal

Symmetry 2019, 11(6), 794; https://doi.org/10.3390/sym11060794 - 14 Jun 2019

Cited by 2 | Viewed by 2545

Abstract

In this paper, we consider a stochastic representation of the epsilon–skew–Cauchy distribution, viewed as a member of the family of skewed distributions discussed in Arellano-Valle et al. (2005). The stochastic representation facilitates derivation of distributional properties of the model. In addition, we introduce [...] Read more.

In this paper, we consider a stochastic representation of the epsilon–skew–Cauchy distribution, viewed as a member of the family of skewed distributions discussed in Arellano-Valle et al. (2005). The stochastic representation facilitates derivation of distributional properties of the model. In addition, we introduce symmetric and asymmetric extensions of the Cauchy distribution, together with an extension of the epsilon–skew–Cauchy distribution. Multivariate versions of these distributions can be envisioned. Bivariate examples are discussed in some detail. Full article

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

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13 pages, 320 KiB

Open AccessArticle

A Note on Ordering Probability Distributions by Skewness

by V. J. García, M. Martel-Escobar and F. J. Vázquez-Polo

Symmetry 2018, 10(7), 286; https://doi.org/10.3390/sym10070286 - 16 Jul 2018

Cited by 3 | Viewed by 2951

Abstract

This paper describes a complementary tool for fitting probabilistic distributions in data analysis. First, we examine the well known bivariate index of skewness and the aggregate skewness function, and then introduce orderings of the skewness of probability distributions. Using an example, we highlight [...] Read more.

This paper describes a complementary tool for fitting probabilistic distributions in data analysis. First, we examine the well known bivariate index of skewness and the aggregate skewness function, and then introduce orderings of the skewness of probability distributions. Using an example, we highlight the advantages of this approach and then present results for these orderings in common uniparametric families of continuous distributions, showing that the orderings are well suited to the intuitive conception of skewness and, moreover, that the skewness can be controlled via the parameter values. Full article

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

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