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19 pages, 738 KiB

Open AccessFeature PaperArticle

A Bivariate Power Lindley Survival Distribution

by Guillermo Martínez-Flórez, Barry C. Arnold and Héctor W. Gómez

Mathematics 2024, 12(21), 3334; https://doi.org/10.3390/math12213334 - 24 Oct 2024

Viewed by 437

Abstract

We introduce and investigate the properties of new families of univariate and bivariate distributions based on the survival function of the Lindley distribution. The univariate distribution, to reflect the nature of its construction, is called a power Lindley survival distribution. The basic distributional [...] Read more.

We introduce and investigate the properties of new families of univariate and bivariate distributions based on the survival function of the Lindley distribution. The univariate distribution, to reflect the nature of its construction, is called a power Lindley survival distribution. The basic distributional properties of this model are described. Maximum likelihood estimates of the parameters of the distribution are studied and the corresponding information matrix is identified. A bivariate power Lindley survival distribution is introduced using the technique of conditional specification. The corresponding joint density and marginal and conditional densities are derived. The product moments of the distribution are obtained, together with bounds on the range of correlations that can be exhibited by the model. Estimation of the parameters of the model is implemented by maximizing the corresponding pseudo-likelihood function. The asymptotic variance–covariance matrix of these estimates is investigated. A simulation study is performed to illustrate the performance of these parameter estimates. Finally some examples of model fitting using real-world data sets are described. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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25 pages, 664 KiB

Open AccessArticle

Extropy and Some of Its More Recent Related Measures for Concomitants of K-Record Values in an Extended FGM Family

by Mohamed A. Abd Elgawad, Haroon M. Barakat, Metwally A. Alawady, Doaa A. Abd El-Rahman, Islam A. Husseiny, Atef F. Hashem and Naif Alotaibi

Mathematics 2023, 11(24), 4934; https://doi.org/10.3390/math11244934 - 12 Dec 2023

Cited by 3 | Viewed by 908

Abstract

This study uses an effective, recently extended Farlie–Gumbel–Morgenstern (EFGM) family to derive the distribution of concomitants of K-record upper values (CKRV). For this CKRV, the negative cumulative residual extropy (NCREX), weighted NCREX (WNCREX), negative cumulative extropy (NCEX), and weighted NCEX (WNCEX) are [...] Read more.

This study uses an effective, recently extended Farlie–Gumbel–Morgenstern (EFGM) family to derive the distribution of concomitants of K-record upper values (CKRV). For this CKRV, the negative cumulative residual extropy (NCREX), weighted NCREX (WNCREX), negative cumulative extropy (NCEX), and weighted NCEX (WNCEX) are theoretically and numerically examined. This study presents several beautiful symmetrical and asymmetric relationships that these inaccuracy measurements satisfy. Additionally, empirical estimations are provided for these measures, and their visualizations enable users to verify their accuracy. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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19 pages, 894 KiB

Open AccessArticle

Bivariate Unit-Weibull Distribution: Properties and Inference

by Roger Tovar-Falón, Guillermo Martínez-Flórez and Luis Páez-Martínez

Mathematics 2023, 11(17), 3760; https://doi.org/10.3390/math11173760 - 1 Sep 2023

Viewed by 1178

Abstract

In this article, we introduce a novel bivariate probability distribution that is absolutely continuous. Considering the Farlie–Gumbel–Morgenstern (FGM) copula and the unit-Weibull distribution, we can obtain a bivariate unit-Weibull distribution. We evaluate the main properties of the new proposal and use two estimation [...] Read more.

In this article, we introduce a novel bivariate probability distribution that is absolutely continuous. Considering the Farlie–Gumbel–Morgenstern (FGM) copula and the unit-Weibull distribution, we can obtain a bivariate unit-Weibull distribution. We evaluate the main properties of the new proposal and use two estimation methods to estimate the parameter for the bivariate probability distribution. A brief Monte Carlo simulation study is conducted to assess the behavior of the employed estimation method and the characteristics of the estimators. Ultimately, as an illustration, a real-life application is presented, demonstrating the utility of the proposal. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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14 pages, 503 KiB

Open AccessArticle

The Log-Bimodal Asymmetric Generalized Gaussian Model with Application to Positive Data

by Guillermo Martínez-Flórez, Roger Tovar-Falón and Heleno Bolfarine

Mathematics 2023, 11(16), 3587; https://doi.org/10.3390/math11163587 - 19 Aug 2023

Viewed by 970

Abstract

One of the most widely known probability distributions used to explain the probabilistic behavior of positive data is the log-normal (LN). Although the LN distribution is capable of adjusting data types, it is not always fully true that the model manages to adequately [...] Read more.

One of the most widely known probability distributions used to explain the probabilistic behavior of positive data is the log-normal (LN). Although the LN distribution is capable of adjusting data types, it is not always fully true that the model manages to adequately model the behavior of the response of interest since in some cases, the degree of skewness and/or kurtosis of the data are greater or less than those that the LN distribution can capture. Another peculiarity of the LN distribution is that it only fits unimodal positive data, which constitutes a limitation when dealing with data that present more than one mode (bimodality). On the other hand, the log-normal model only fits unimodal positive data and in reality there are multiple applications where the behavior of materials is bimodal. To fill this gap, this paper introduces a new probability distribution that is capable of fitting unimodal or bimodal positive data with a high or low degree of skewness and/or kurtosis. The new distribution is a generalization of the LN distribution. For the new proposal, its main properties are studied and the process of estimation of the parameters involved in the model is carried out from a classical perspective using the maximum likelihood method. An important feature of this distribution is the non-singularity of the Fisher information matrix, which guarantees the use of asymptotic theory to study the properties of the parameter estimators. A Monte Carlo type simulation study is carried out to evaluate the properties of the estimators and finally, an illustration is presented with a set of data related to the concentration of nickel in soil samples, allowing to show that the proposed distribution fits extremely well in certain situations. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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22 pages, 1117 KiB

Open AccessArticle

The NBRULC Reliability Class: Mathematical Theory and Goodness-of-Fit Testing with Applications to Asymmetric Censored and Uncensored Data

by Walid B. H. Etman, Mohamed S. Eliwa, Hana N. Alqifari, Mahmoud El-Morshedy, Laila A. Al-Essa and Rashad M. EL-Sagheer

Mathematics 2023, 11(13), 2805; https://doi.org/10.3390/math11132805 - 21 Jun 2023

Cited by 1 | Viewed by 1073

Abstract

The majority of approaches proposed in the past few decades to solve life test problems have differed markedly from those used for closely related, yet broader, issues. Due to the complexity of data that are generated each day in many practical domains, as [...] Read more.

The majority of approaches proposed in the past few decades to solve life test problems have differed markedly from those used for closely related, yet broader, issues. Due to the complexity of data that are generated each day in many practical domains, as a result of the development of scales for rating the success or failure of reliability, a new domain of reliability has been created. This domain is referred to as life classes, where specific probability distributions are presented. In this study, it is shown that the use of the quality-of-fit technique to solve problems involving life testing makes sense, and produces simpler processes that are roughly equivalent or superior to those used in traditional procedures. They may also behave better in limited samples. This work investigates a novel quality-of-fit test statistic; it is based on an exponential transform and is compared to the best renewal used Laplace test in increasing convex ordering (NBRULC). Evidence for approach normality is provided. The calculated variables include powers, Pitman asymptotic effectiveness, and critical points. Methods on how to handle censored data were also studied. Our experiments have real-world applications in the fields of medicine and engineering. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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14 pages, 581 KiB

Open AccessArticle

Reconstructing the Quarterly Series of the Chilean Gross Domestic Product Using a State Space Approach

by Christian Caamaño-Carrillo, Sergio Contreras-Espinoza and Orietta Nicolis

Mathematics 2023, 11(8), 1827; https://doi.org/10.3390/math11081827 - 12 Apr 2023

Viewed by 1215

Abstract

In this work, we use a cointegration state space approach to estimate the quarterly series of the Chilean Gross Domestic Product (GDP) in the period 1965–2009. First, the equation of Engle–Granger is estimated using the data of the yearly GPD and some related [...] Read more.

In this work, we use a cointegration state space approach to estimate the quarterly series of the Chilean Gross Domestic Product (GDP) in the period 1965–2009. First, the equation of Engle–Granger is estimated using the data of the yearly GPD and some related variables, such as the price of copper, the exports of goods and services, and the mining production index. The estimated coefficients of this regression are then used to obtain a first estimation of the quarterly GDP series with measurement errors. A state space model is then applied to improve the preliminary estimation of the GDP with the main purpose of eliminating the maximum error of measurement such that the sum of the three-month values coincide swith the yearly GDP. The results are then compared with the traditional disaggregation methods. The resulting quarterly GDP series reflects the major events of the historical Chilean economy. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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19 pages, 464 KiB

Open AccessArticle

Some Extensions of the Asymmetric Exponentiated Bimodal Normal Model for Modeling Data with Positive Support

by Roger Tovar-Falón, Guillermo Martínez-Flórez and Isaías Ceña-Tapia

Mathematics 2023, 11(7), 1563; https://doi.org/10.3390/math11071563 - 23 Mar 2023

Cited by 1 | Viewed by 1294

Abstract

It is common in many fields of knowledge to assume that the data under study have a normal distribution, which often generates mistakes in the results, since this assumption does not always coincide with the characteristics of the observations under analysis. In some [...] Read more.

It is common in many fields of knowledge to assume that the data under study have a normal distribution, which often generates mistakes in the results, since this assumption does not always coincide with the characteristics of the observations under analysis. In some cases, the data may have degrees of skewness and/or kurtosis greater than what the normal model can capture, and in others, they may present two or more modes. In this work, two new families of skewed distributions are presented that fit bimodal data with positive support. The new families were obtained from the extension of the bimodal normal distribution to the alpha-power family class. The proposed distributions were studied for their main properties, such as their probability density function, cumulative distribution function, survival function, and hazard function. The parameter estimation process was performed from a classical perspective using the maximum likelihood method. The non-singularity of Fisher’s information was demonstrated, which made it possible to find the stochastic convergence of the vector of the maximum likelihood estimators and, based on the latter, perform statistical inference via the likelihood ratio. The applicability of the proposed distributions was exemplified using real data sets. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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21 pages, 1249 KiB

Open AccessArticle

The Multivariate Skewed Log-Birnbaum–Saunders Distribution and Its Associated Regression Model

by Guillermo Martínez-Flórez, Sandra Vergara-Cardozo, Roger Tovar-Falón and Luisa Rodriguez-Quevedo

Mathematics 2023, 11(5), 1095; https://doi.org/10.3390/math11051095 - 22 Feb 2023

Cited by 1 | Viewed by 1677

Abstract

In this article, a multivariate extension of the unit-sinh-normal (USHN) distribution is presented. The new distribution, which is obtained from the conditionally specified distributions methodology, is absolutely continuous, and its marginal distributions are univariate USHN. The properties of the multivariate USHN distribution are [...] Read more.

In this article, a multivariate extension of the unit-sinh-normal (USHN) distribution is presented. The new distribution, which is obtained from the conditionally specified distributions methodology, is absolutely continuous, and its marginal distributions are univariate USHN. The properties of the multivariate USHN distribution are studied in detail, and statistical inference is carried out from a classical approach using the maximum likelihood method. The new multivariate USHN distribution is suitable for modeling bounded data, especially in the

{(0, 1)}^{p}

region. In addition, the proposed distribution is extended to the case of the regression model and, for the latter, the Fisher information matrix is derived. The numerical results of a small simulation study and two applications with real data sets allow us to conclude that the proposed distribution, as well as its extension to regression models, are potentially useful to analyze the data of proportions, rates, or indices when modeling them jointly considering different degrees of correlation that may exist in the study variables is of interest. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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16 pages, 445 KiB

Open AccessArticle

The Slash Half-Normal Distribution Applied to a Cure Rate Model with Application to Bone Marrow Transplantation

by Diego I. Gallardo, Yolanda M. Gómez, Héctor J. Gómez, María José Gallardo-Nelson and Marcelo Bourguignon

Mathematics 2023, 11(3), 518; https://doi.org/10.3390/math11030518 - 18 Jan 2023

Cited by 1 | Viewed by 1428

Abstract

This paper proposes, for the first time, the use of an asymmetric positive and heavy-tailed distribution in a cure rate model context. In particular, it introduces a cure-rate survival model by assuming that the time-to-event of interest follows a slash half-normal distribution and [...] Read more.

This paper proposes, for the first time, the use of an asymmetric positive and heavy-tailed distribution in a cure rate model context. In particular, it introduces a cure-rate survival model by assuming that the time-to-event of interest follows a slash half-normal distribution and that the number of competing causes of the event of interest follows a power series distribution, which defines six new cure rate models. Several properties of the model are derived and an alternative expression for the cumulative distribution function of the model is presented, which is very useful for the computational implementation of the model. A procedure based on the expectation–maximization algorithm is proposed for the parameter estimation. Two simulation studies are performed to assess some properties of the estimators, showing the good performance of the proposed estimators in finite samples. Finally, an application to a bone marrow transplant data set is presented. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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17 pages, 358 KiB

Open AccessArticle

COVID-19 Active Case Forecasts in Latin American Countries Using Score-Driven Models

by Sergio Contreras-Espinoza, Francisco Novoa-Muñoz, Szabolcs Blazsek, Pedro Vidal and Christian Caamaño-Carrillo

Mathematics 2023, 11(1), 136; https://doi.org/10.3390/math11010136 - 27 Dec 2022

Viewed by 1610

Abstract

With the aim of mitigating the damage caused by the coronavirus disease 2019 (COVID-19) pandemic, it is important to use models that allow forecasting possible new infections accurately in order to face the pandemic in specific sociocultural contexts in the best possible way. [...] Read more.

With the aim of mitigating the damage caused by the coronavirus disease 2019 (COVID-19) pandemic, it is important to use models that allow forecasting possible new infections accurately in order to face the pandemic in specific sociocultural contexts in the best possible way. Our first contribution is empirical. We use an extensive COVID-19 dataset from nine Latin American countries for the period of 1 April 2020 to 31 December 2021. Our second and third contributions are methodological. We extend relevant (i) state-space models with score-driven dynamics and (ii) nonlinear state-space models with unobserved components, respectively. We use weekly seasonal effects, in addition to the local-level and trend filters of the literature, for (i) and (ii), and the negative binomial distribution for (ii). We find that the statistical and forecasting performances of the novel score-driven specifications are superior to those of the nonlinear state-space models with unobserved components model, providing a potential valid alternative to forecasting the number of possible new COVID-19 infections. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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18 pages, 731 KiB

Open AccessArticle

Power Families of Bivariate Proportional Hazard Models

by Guillermo Martínez-Flórez, Carlos Barrera-Causil and Artur J. Lemonte

Mathematics 2022, 10(23), 4410; https://doi.org/10.3390/math10234410 - 23 Nov 2022

Viewed by 1403

Abstract

In this paper, we propose a general class of bivariate proportional hazard distributions, which is based on the family of asymmetric proportional hazard distributions and the bivariate Pareto copula. Distributional properties of the bivariate proportional hazard distribution are derived. We specialize the bivariate [...] Read more.

In this paper, we propose a general class of bivariate proportional hazard distributions, which is based on the family of asymmetric proportional hazard distributions and the bivariate Pareto copula. Distributional properties of the bivariate proportional hazard distribution are derived. We specialize the bivariate proportional hazard family of distributions to the normal case, and so we introduce the bivariate proportional hazard normal distribution. Parameter estimation by the maximum likelihood method of the bivariate proportional hazard normal distribution is then discussed. Finally, an application of the new bivariate distribution to real data is considered for illustrative purposes. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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14 pages, 436 KiB

Open AccessArticle

An Alternative to the Log-Skew-Normal Distribution: Properties, Inference, and an Application to Air Pollutant Concentrations

by Jaime Arrué, Reinaldo Boris Arellano-Valle, Osvaldo Venegas, Heleno Bolfarine and Héctor W. Gómez

Mathematics 2022, 10(22), 4336; https://doi.org/10.3390/math10224336 - 18 Nov 2022

Viewed by 1318

Abstract

In this study, we consider an alternative to the log-skew-normal distribution. It is called the modified log-skew-normal distribution and introduces greater flexibility in the skewness and kurtosis parameters. We first study several of the main probabilistic properties of the new distribution, such as [...] Read more.

In this study, we consider an alternative to the log-skew-normal distribution. It is called the modified log-skew-normal distribution and introduces greater flexibility in the skewness and kurtosis parameters. We first study several of the main probabilistic properties of the new distribution, such as the computation of its moments and the non-existence of the moment-generating function. Parameter estimation by the maximum likelihood approach is also studied. This approach presents an overestimation problem in the shape parameter, which in some cases, can even be infinite. However, as we demonstrate, this problem is solved by adapting bias reduction using Firth’s approach. We also show that the modified log-skew-normal model likewise inherits the non-singularity of the Fisher information matrix of the untransformed model, when the shape parameter is null. Finally, we apply the modified log-skew-normal model to a real example related to pollution data. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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Review

Jump to: Research

19 pages, 461 KiB

Open AccessReview

An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations

by Yolanda M. Gómez, Diego I. Gallardo, Carolina Marchant, Luis Sánchez and Marcelo Bourguignon

Mathematics 2024, 12(1), 56; https://doi.org/10.3390/math12010056 - 23 Dec 2023

Cited by 3 | Viewed by 5079

Abstract

The Weibull distribution is a versatile probability distribution widely applied in modeling the failure times of objects or systems. Its behavior is shaped by two essential parameters: the shape parameter and the scale parameter. By manipulating these parameters, the Weibull distribution adeptly captures [...] Read more.

The Weibull distribution is a versatile probability distribution widely applied in modeling the failure times of objects or systems. Its behavior is shaped by two essential parameters: the shape parameter and the scale parameter. By manipulating these parameters, the Weibull distribution adeptly captures diverse failure patterns observed in real-world scenarios. This flexibility and broad applicability make it an indispensable tool in reliability analysis and survival modeling. This manuscript explores five parameterizations of the Weibull distribution, each based on different moments, like mean, quantile, and mode. It meticulously characterizes each parameterization, introducing a novel one based on the model’s mode, along with its hazard and survival functions, shedding light on their unique properties. Additionally, it delves into the interpretation of regression coefficients when incorporating regression structures into these parameterizations. It is analytically established that all five parameterizations define the same log-likelihood function, underlining their equivalence. Through Monte Carlo simulation studies, the performances of these parameterizations are evaluated in terms of parameter estimations and residuals. The models are further applied to real-world data, illustrating their effectiveness in analyzing material fatigue life and survival data. In summary, this manuscript provides a comprehensive exploration of the Weibull distribution and its various parameterizations. It offers valuable insights into their applications and implications in modeling failure times, with potential contributions to diverse fields requiring reliability and survival analysis. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry)

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