Symmetry

Research

15 pages, 1528 KiB

Open AccessArticle

SimBetaReg Web-Tool: The Easiest Way to Implement the Beta and Simplex Regression Models

by Emrah Altun and Mahmoud El-Morshedy

Symmetry 2021, 13(12), 2437; https://doi.org/10.3390/sym13122437 - 16 Dec 2021

Cited by 2 | Viewed by 2613

When the response variable is defined on the (0,1) interval, the beta and simplex regression models are commonly used by researchers. However, there is no software support for these models to make their implementation easy for researchers. In this study, we developed a [...] Read more.

When the response variable is defined on the (0,1) interval, the beta and simplex regression models are commonly used by researchers. However, there is no software support for these models to make their implementation easy for researchers. In this study, we developed a web-tool, named SimBetaReg, to help researchers who are not familiar with programming to implement the beta and simplex regression models. The developed application is free and works independently from the operating systems. Additionally, we model the incidence ratios of COVID-19 with educational and civic engagement indicators of the OECD countries using the SimBetaReg web-tool. Empirical findings show that when the educational attainment, years in education, and voter turnout increase, the incidence ratios of the countries decrease. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

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23 pages, 1199 KiB

Open AccessArticle

The Multi-Compartment SI(RD) Model with Regime Switching: An Application to COVID-19 Pandemic

by Manuel L. Esquível, Nadezhda P. Krasii, Gracinda R. Guerreiro and Paula Patrício

Symmetry 2021, 13(12), 2427; https://doi.org/10.3390/sym13122427 - 15 Dec 2021

Cited by 6 | Viewed by 2569

Abstract

We study—with existence and unicity results—a variant of the SIR model for an infectious disease incorporating both the possibility of a death outcome—in a short period of time—and a regime switch that can account for the mitigation measures used to control the spreading [...] Read more.

We study—with existence and unicity results—a variant of the SIR model for an infectious disease incorporating both the possibility of a death outcome—in a short period of time—and a regime switch that can account for the mitigation measures used to control the spreading of the infections, such as a total lockdown. This model is parametrised by three parameters: the basic reproduction number, the mortality rate of the infected, and the duration of the disease. We discuss a particular example of application to Portuguese COVID-19 data in two short periods just after the start of the epidemic in 4 March 2020, with the first two cases dated that day. We propose a simple and effective method for the estimation of the main parameters of the disease, namely, the basic reproduction number and the mortality rate of the infected. We correct these estimated values to take into account the asymptomatic non-diagnosed members of the population. We compare the outcome of the model in the cases of the existence, or not, of a regime switch, and under three different scenarios, with a remarkable agreement between model and data deaths in the case of our basis scenario. In a final short remark, we deal with the existence of symmetries for the proposed model. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

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15 pages, 309 KiB

Open AccessArticle

Efficient Estimation for the Derivative of Nonparametric Function by Optimally Combining Quantile Information

by Xiaoshuang Zhou, Xiulian Gao, Yukun Zhang, Xiuling Yin and Yanfeng Shen

Symmetry 2021, 13(12), 2387; https://doi.org/10.3390/sym13122387 - 10 Dec 2021

Cited by 1 | Viewed by 1968

Abstract

In this article, we focus on the efficient estimators of the derivative of the nonparametric function in the nonparametric quantile regression model. We develop two ways of combining quantile regression information to derive the estimators. One is the weighted composite quantile regression estimator [...] Read more.

In this article, we focus on the efficient estimators of the derivative of the nonparametric function in the nonparametric quantile regression model. We develop two ways of combining quantile regression information to derive the estimators. One is the weighted composite quantile regression estimator based on the quantile weighted loss function; the other is the weighted quantile average estimator based on the weighted average of quantile regression estimators at a single quantile. Furthermore, by minimizing the asymptotic variance, the optimal weight vector is computed, and consequently, the optimal estimator is obtained. Furthermore, we conduct some simulations to evaluate the performance of our proposed estimators under different symmetric error distributions. Simulation studies further illustrate that both estimators work better than the local linear least square estimator for all the symmetric errors considered except the normal error, and the weighted quantile average estimator performs better than the weighted composite quantile regression estimator in most situations. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

17 pages, 360 KiB

Open AccessArticle

Asymptotic Results for Multinomial Models

by Isaac Akoto, João T. Mexia and Filipe J. Marques

Symmetry 2021, 13(11), 2173; https://doi.org/10.3390/sym13112173 - 12 Nov 2021

Cited by 2 | Viewed by 1650

Abstract

In this work, we derived new asymptotic results for multinomial models. To obtain these results, we started by studying limit distributions in models with a compact parameter space. This restriction holds since the key parameter whose components are the probabilities of the possible [...] Read more.

In this work, we derived new asymptotic results for multinomial models. To obtain these results, we started by studying limit distributions in models with a compact parameter space. This restriction holds since the key parameter whose components are the probabilities of the possible outcomes have non-negative components that add up to 1. Based on these results, we obtained confidence ellipsoids and simultaneous confidence intervals for models with normal limit distributions. We then studied the covariance matrices of the limit normal distributions for the multinomial models. This was a transition between the previous general results and on the inference for multinomial models in which we considered the chi-square tests, confidence regions and non-linear statistics—namely log-linear models with two numerical applications to those models. Namely, our approach overcame the hierarchical restrictions assumed to analyse the multidimensional contingency table. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

15 pages, 320 KiB

Open AccessArticle

Bayesian Estimation for the Coefficients of Variation of Birnbaum–Saunders Distributions

by Wisunee Puggard, Sa-Aat Niwitpong and Suparat Niwitpong

Symmetry 2021, 13(11), 2130; https://doi.org/10.3390/sym13112130 - 9 Nov 2021

Cited by 6 | Viewed by 2002

Abstract

The Birnbaum–Saunders (BS) distribution, which is asymmetric with non-negative support, can be transformed to a normal distribution, which is symmetric. Therefore, the BS distribution is useful for describing data comprising values greater than zero. The coefficient of variation (CV), which is an important [...] Read more.

The Birnbaum–Saunders (BS) distribution, which is asymmetric with non-negative support, can be transformed to a normal distribution, which is symmetric. Therefore, the BS distribution is useful for describing data comprising values greater than zero. The coefficient of variation (CV), which is an important descriptive statistic for explaining variation within a dataset, has not previously been used for statistical inference on a BS distribution. The aim of this study is to present four methods for constructing confidence intervals for the CV, and the difference between the CVs of BS distributions. The proposed methods are based on the generalized confidence interval (GCI), a bootstrapped confidence interval (BCI), a Bayesian credible interval (BayCI), and the highest posterior density (HPD) interval. A Monte Carlo simulation study was conducted to evaluate their performances in terms of coverage probability and average length. The results indicate that the HPD interval was the best-performing method overall. PM 2.5 concentration data for Chiang Mai, Thailand, collected in March and April 2019, were used to illustrate the efficacies of the proposed methods, the results of which were in good agreement with the simulation study findings. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

14 pages, 521 KiB

Open AccessArticle

New Sufficient Conditions to Ulam Stabilities for a Class of Higher Order Integro-Differential Equations

by Alberto M. Simões, Fernando Carapau and Paulo Correia

Symmetry 2021, 13(11), 2068; https://doi.org/10.3390/sym13112068 - 2 Nov 2021

Cited by 4 | Viewed by 1463

Abstract

In this work, we present sufficient conditions in order to establish different types of Ulam stabilities for a class of higher order integro-differential equations. In particular, we consider a new kind of stability, the

σ

-semi-Hyers-Ulam stability, which is in some sense between [...] Read more.

In this work, we present sufficient conditions in order to establish different types of Ulam stabilities for a class of higher order integro-differential equations. In particular, we consider a new kind of stability, the

σ

-semi-Hyers-Ulam stability, which is in some sense between the Hyers–Ulam and the Hyers–Ulam–Rassias stabilities. These new sufficient conditions result from the application of the Banach Fixed Point Theorem, and by applying a specific generalization of the Bielecki metric. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

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26 pages, 740 KiB

Open AccessArticle

PQMLE of a Partially Linear Varying Coefficient Spatial Autoregressive Panel Model with Random Effects

by Shuangshuang Li, Jianbao Chen and Danqing Chen

Symmetry 2021, 13(11), 2057; https://doi.org/10.3390/sym13112057 - 1 Nov 2021

Cited by 2 | Viewed by 1449

Abstract

This article deals with asymmetrical spatial data which can be modeled by a partially linear varying coefficient spatial autoregressive panel model (PLVCSARPM) with random effects. We constructed its profile quasi-maximum likelihood estimators (PQMLE). The consistency and asymptotic normality of the estimators were proved [...] Read more.

This article deals with asymmetrical spatial data which can be modeled by a partially linear varying coefficient spatial autoregressive panel model (PLVCSARPM) with random effects. We constructed its profile quasi-maximum likelihood estimators (PQMLE). The consistency and asymptotic normality of the estimators were proved under some regular conditions. Monte Carlo simulations implied our estimators have good finite sample performance. Finally, a set of asymmetric real data applications was analyzed for illustrating the performance of the provided method. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

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20 pages, 1797 KiB

Open AccessArticle

Modeling Extreme Values Utilizing an Asymmetric Probability Function

by Mohammed M. A. Almazah, Muqrin A. Almuqrin, Mohamed. S. Eliwa, Mahmoud El-Morshedy and Haitham M. Yousof

Symmetry 2021, 13(9), 1730; https://doi.org/10.3390/sym13091730 - 18 Sep 2021

Cited by 14 | Viewed by 1798

Abstract

In this article, a new flexible probability density function with three parameters is proposed for modeling asymmetric data (positive and negative) with different types of kurtosis (mesokurtic, leptokurtic and platykurtic). Some of its statistical and reliability properties, including hazard rate function, moments, moment [...] Read more.

In this article, a new flexible probability density function with three parameters is proposed for modeling asymmetric data (positive and negative) with different types of kurtosis (mesokurtic, leptokurtic and platykurtic). Some of its statistical and reliability properties, including hazard rate function, moments, moment generating function, incomplete moments, mean deviations, moment of the residual life, moment of the reversed residual life, and order statistics are derived. Its hazard rate function can be either constant, increasing-constant, decreasing-constant, U shape, upside down shape or upside down-U shape. Seven classical estimation methods are considered to estimate the unknown model parameters. Monte Carlo simulation experiments are performed to compare the performance of the seven different estimation methods. Finally, a distinctive asymmetric real data application is analyzed for illustrating the flexibility of the new model. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

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21 pages, 2881 KiB

Open AccessArticle

Bayesian Analysis of Partially Linear Additive Spatial Autoregressive Models with Free-Knot Splines

by Zhiyong Chen and Jianbao Chen

Symmetry 2021, 13(9), 1635; https://doi.org/10.3390/sym13091635 - 6 Sep 2021

Cited by 4 | Viewed by 2036

Abstract

This article deals with symmetrical data that can be modelled based on Gaussian distribution. We consider a class of partially linear additive spatial autoregressive (PLASAR) models for spatial data. We develop a Bayesian free-knot splines approach to approximate the nonparametric functions. It can [...] Read more.

This article deals with symmetrical data that can be modelled based on Gaussian distribution. We consider a class of partially linear additive spatial autoregressive (PLASAR) models for spatial data. We develop a Bayesian free-knot splines approach to approximate the nonparametric functions. It can be performed to facilitate efficient Markov chain Monte Carlo (MCMC) tools to design a Gibbs sampler to explore the full conditional posterior distributions and analyze the PLASAR models. In order to acquire a rapidly-convergent algorithm, a modified Bayesian free-knot splines approach incorporated with powerful MCMC techniques is employed. The Bayesian estimator (BE) method is more computationally efficient than the generalized method of moments estimator (GMME) and thus capable of handling large scales of spatial data. The performance of the PLASAR model and methodology is illustrated by a simulation, and the model is used to analyze a Sydney real estate dataset. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

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22 pages, 817 KiB

Open AccessArticle

Estimation and Prediction for Nadarajah-Haghighi Distribution under Progressive Type-II Censoring

by Mingjie Wu and Wenhao Gui

Symmetry 2021, 13(6), 999; https://doi.org/10.3390/sym13060999 - 3 Jun 2021

Cited by 15 | Viewed by 2467

Abstract

The paper discusses the estimation and prediction problems for the Nadarajah-Haghighi distribution using progressive type-II censored samples. For the unknown parameters, we first calculate the maximum likelihood estimates through the Expectation–Maximization algorithm. In order to choose the best Bayesian estimator, a loss function [...] Read more.

The paper discusses the estimation and prediction problems for the Nadarajah-Haghighi distribution using progressive type-II censored samples. For the unknown parameters, we first calculate the maximum likelihood estimates through the Expectation–Maximization algorithm. In order to choose the best Bayesian estimator, a loss function must be specified. When the loss is essentially symmetric, it is reasonable to use the square error loss function. However, for some estimation problems, the actual loss is often asymmetric. Therefore, we also need to choose an asymmetric loss function. Under the balanced squared error and symmetric squared error loss functions, the Tierney and Kadane method is used for calculating different kinds of approximate Bayesian estimates. The Metropolis-Hasting algorithm is also provided here. In addition, we construct a variety of interval estimations of the unknown parameters including asymptotic intervals, bootstrap intervals, and highest posterior density intervals using the sample derived from the Metropolis-Hasting algorithm. Furthermore, we compute the point predictions and predictive intervals for a future sample when facing the one-sample and two-sample situations. At last, we compare and appraise the performance of the provided techniques by carrying out a simulation study and analyzing a real rainfall data set. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

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20 pages, 317 KiB

Open AccessArticle

Estimation and Prediction for Gompertz Distribution under General Progressive Censoring

by Yuxuan Wang and Wenhao Gui

Symmetry 2021, 13(5), 858; https://doi.org/10.3390/sym13050858 - 12 May 2021

Cited by 4 | Viewed by 2283

Abstract

In this article, we discuss the estimation of the parameters for Gompertz distribution and prediction using general progressive Type-II censoring. Based on the Expectation–Maximization algorithm, we calculate the maximum likelihood estimates. Bayesian estimates are considered under different loss functions, which are symmetrical, asymmetrical [...] Read more.

In this article, we discuss the estimation of the parameters for Gompertz distribution and prediction using general progressive Type-II censoring. Based on the Expectation–Maximization algorithm, we calculate the maximum likelihood estimates. Bayesian estimates are considered under different loss functions, which are symmetrical, asymmetrical and balanced, respectively. An approximate method—Tierney and Kadane—is used to derive the estimates. Besides, the Metropolis-Hasting (MH) algorithm is applied to get the Bayesian estimates as well. According to Fisher information matrix, we acquire asymptotic confidence intervals. Bootstrap intervals are also established. Furthermore, we build the highest posterior density intervals through the sample generated by the MH algorithm. Then, Bayesian predictive intervals and estimates for future samples are provided. Finally, for evaluating the quality of the approaches, a numerical simulation study is implemented. In addition, we analyze two real datasets. Full article

(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)

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