scholarly journals On the Regression Model for Generalized Normal Distributions

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 173
Author(s):  
Ayman Alzaatreh ◽  
Mohammad Aljarrah ◽  
Ayanna Almagambetova ◽  
Nazgul Zakiyeva
Keyword(s):  
Maximum Likelihood ◽  
Regression Model ◽  
Normal Distribution ◽  
Regression Models ◽  
Likelihood Method ◽  
Normal Distributions ◽  
Normality Assumption ◽  
Normal Regression ◽  
Modeling Data ◽  
Model Residuals

The traditional linear regression model that assumes normal residuals is applied extensively in engineering and science. However, the normality assumption of the model residuals is often ineffective. This drawback can be overcome by using a generalized normal regression model that assumes a non-normal response. In this paper, we propose regression models based on generalizations of the normal distribution. The proposed regression models can be used effectively in modeling data with a highly skewed response. Furthermore, we study in some details the structural properties of the proposed generalizations of the normal distribution. The maximum likelihood method is used for estimating the parameters of the proposed method. The performance of the maximum likelihood estimators in estimating the distributional parameters is assessed through a small simulation study. Applications to two real datasets are given to illustrate the flexibility and the usefulness of the proposed distributions and their regression models.

The Odd Log-Logistic Geometric Normal Regression Model with Applications

2019 ◽  
Vol 11 (01n02) ◽  
pp. 1950003
Author(s):  
Fábio Prataviera ◽  
Gauss M. Cordeiro ◽  
Edwin M. M. Ortega ◽  
Adriano K. Suzuki
Keyword(s):  
Regression Model ◽  
Normal Distribution ◽  
Regression Models ◽  
Empirical Distribution ◽  
Real Data ◽  
Likelihood Method ◽  
Standard Normal Distribution ◽  
Model Parameters ◽  
Diagnostic Measures ◽  
Standard Normal

In several applications, the distribution of the data is frequently unimodal, asymmetric or bimodal. The regression models commonly used for applications to data with real support are the normal, skew normal, beta normal and gamma normal, among others. We define a new regression model based on the odd log-logistic geometric normal distribution for modeling asymmetric or bimodal data with support in [Formula: see text], which generalizes some known regression models including the widely known heteroscedastic linear regression. We adopt the maximum likelihood method for estimating the model parameters and define diagnostic measures to detect influential observations. For some parameter settings, sample sizes and different systematic structures, various simulations are performed to verify the adequacy of the estimators of the model parameters. The empirical distribution of the quantile residuals is investigated and compared with the standard normal distribution. We prove empirically the usefulness of the proposed models by means of three applications to real data.

scholarly journals Statistical tests for latent class in censored data due to detection limit

2019 ◽  
Vol 29 (8) ◽  
pp. 2179-2197
Author(s):  
Hua He ◽  
Wan Tang ◽  
Tanika Kelly ◽  
Shengxu Li ◽  
Jiang He
Keyword(s):  
Regression Model ◽  
Normal Distribution ◽  
Regression Models ◽  
Latent Class ◽  
Limit Of Detection ◽  
Statistical Tests ◽  
Ratio Test ◽  
Tobit Regression ◽  
Normal Regression ◽  
Concentration Levels

Measures of substance concentration in urine, serum or other biological matrices often have an assay limit of detection. When concentration levels fall below the limit, the exact measures cannot be obtained. Instead, the measures are censored as only partial information that the levels are under the limit is known. Assuming the concentration levels are from a single population with a normal distribution or follow a normal distribution after some transformation, Tobit regression models, or censored normal regression models, are the standard approach for analyzing such data. However, in practice, it is often the case that the data can exhibit more censored observations than what would be expected under the Tobit regression models. One common cause is the heterogeneity of the study population, caused by the existence of a latent group of subjects who lack the substance measured. For such subjects, the measurements will always be under the limit. If a censored normal regression model is appropriate for modeling the subjects with the substance, the whole population follows a mixture of a censored normal regression model and a degenerate distribution of the latent class. While there are some studies on such mixture models, a fundamental question about testing whether such mixture modeling is necessary, i.e. whether such a latent class exists, has not been studied yet. In this paper, three tests including Wald test, likelihood ratio test and score test are developed for testing the existence of such latent class. Simulation studies are conducted to evaluate the performance of the tests, and two real data examples are employed to illustrate the tests.

scholarly journals New Regression Models Based on the Unit-Sinh-Normal Distribution: Properties, Inference, and Applications

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1231
Author(s):  
Guillermo Martínez-Flórez ◽  
Roger Tovar-Falón
Keyword(s):  
Maximum Likelihood ◽  
Regression Models ◽  
Maximum Likelihood Method ◽  
Real Data ◽  
Random Variable ◽  
Maximum Likelihood Estimates ◽  
Likelihood Method ◽  
Data Sets ◽  
Modeling Data ◽  
Positive Support

In this paper, two new distributions were introduced to model unimodal and/or bimodal data. The first distribution, which was obtained by applying a simple transformation to a unit-Birnbaum–Saunders random variable, is useful for modeling data with positive support, while the second is appropriate for fitting data on the (0,1) interval. Extensions to regression models were also studied in this work, and statistical inference was performed from a classical perspective by using the maximum likelihood method. A small simulation study is presented to evaluate the benefits of the maximum likelihood estimates of the parameters. Finally, two applications to real data sets are reported to illustrate the developed methodology.

scholarly journals Finite mixture of compositional regression with gaussian errors

2018 ◽  
Vol 41 (1) ◽  
pp. 75-86
Author(s):  
Taciana Shimizu ◽  
Francisco Louzada ◽  
Adriano Suzuki
Keyword(s):  
Maximum Likelihood ◽  
Em Algorithm ◽  
Regression Model ◽  
Simulation Study ◽  
Regression Models ◽  
Finite Mixture ◽  
Maximum Likelihood Estimates ◽  
The Real ◽  
Compositional Structure ◽  
Volleyball Players

In this paper, we consider to evaluate the efficiency of volleyball players according to the performance of attack, block and serve, but considering the compositional structure of the data related to the fundaments. The finite mixture of regression models better fitted the data in comparison with the usual regression model. The maximum likelihood estimates are obtained via an EM algorithm. A simulation study revels that the estimates are closer to the real values, the estimators are asymptotically unbiased for the parameters. A real Brazilian volleyball dataset related to the efficiency of the players is considered for the analysis.

scholarly journals Using Iterative Reweighting Algorithm and Genetic Algorithm to Calculate The Estimation of The Parameters Of The Maximum Likelihood of The Skew Normal Distribution

2021 ◽  
Vol 27 (127) ◽  
pp. 253-264
Author(s):  
مرتضى علاء الخفاجي ◽  
رباب عبد الرضا البكري
Keyword(s):  
Genetic Algorithm ◽  
Maximum Likelihood ◽  
Normal Distribution ◽  
Likelihood Estimation ◽  
Real Data ◽  
Likelihood Method ◽  
Skew Normal Distribution ◽  
Iterative Reweighting ◽  
Non Linear Equation ◽  
Skew Normal

Excessive skewness which occurs sometimes in the data is represented as an obstacle against normal distribution. So, recent studies have witnessed activity in studying the skew-normal distribution (SND) that matches the skewness data which is regarded as a special case of the normal distribution with additional skewness parameter (α), which gives more flexibility to the normal distribution. When estimating the parameters of (SND), we face the problem of the non-linear equation and by using the method of Maximum Likelihood estimation (ML) their solutions will be inaccurate and unreliable. To solve this problem, two methods can be used that are: the genetic algorithm (GA) and the iterative reweighting algorithm (IR) based on the Maximum Likelihood method. Monte Carlo simulation was used with different skewness levels and sample sizes, and the superiority of the results was compared. It was concluded that (SND) model estimation using (GA) is the best when the samples sizes are small and medium, while large samples indicate that the (IR) algorithm is the best. The study was also done using real data to find the parameter estimation and a comparison between the superiority of the results based on (AIC, BIC, Mse and Def) criteria.

scholarly journals A New Quantile Regression Model and Its Diagnostic Analytics for a Weibull Distributed Response with Applications

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2768
Author(s):  
Luis Sánchez ◽  
Víctor Leiva ◽  
Helton Saulo ◽  
Carolina Marchant ◽  
José M. Sarabia
Keyword(s):  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Quantile Regression ◽  
Regression Model ◽  
Likelihood Method ◽  
Model Parameters ◽  
Distributed Data ◽  
Quantile Regression Model ◽  
The Mean ◽  
Asymmetrically Distributed

Standard regression models focus on the mean response based on covariates. Quantile regression describes the quantile for a response conditioned to values of covariates. The relevance of quantile regression is even greater when the response follows an asymmetrical distribution. This relevance is because the mean is not a good centrality measure to resume asymmetrically distributed data. In such a scenario, the median is a better measure of the central tendency. Quantile regression, which includes median modeling, is a better alternative to describe asymmetrically distributed data. The Weibull distribution is asymmetrical, has positive support, and has been extensively studied. In this work, we propose a new approach to quantile regression based on the Weibull distribution parameterized by its quantiles. We estimate the model parameters using the maximum likelihood method, discuss their asymptotic properties, and develop hypothesis tests. Two types of residuals are presented to evaluate the model fitting to data. We conduct Monte Carlo simulations to assess the performance of the maximum likelihood estimators and residuals. Local influence techniques are also derived to analyze the impact of perturbations on the estimated parameters, allowing us to detect potentially influential observations. We apply the obtained results to a real-world data set to show how helpful this type of quantile regression model is.

scholarly journals On Properties of the Bimodal Skew-Normal Distribution and an Application

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 703
Author(s):  
David Elal-Olivero ◽  
Juan F. Olivares-Pacheco ◽  
Osvaldo Venegas ◽  
Heleno Bolfarine ◽  
Héctor W. Gómez
Keyword(s):  
Maximum Likelihood ◽  
Normal Distribution ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Real Data ◽  
Skew Normal Distribution ◽  
Data Set ◽  
Normal Distributions ◽  
Stochastic Representations ◽  
Skew Normal

The main object of this paper is to develop an alternative construction for the bimodal skew-normal distribution. The construction is based upon a study of the mixture of skew-normal distributions. We study some basic properties of this family, its stochastic representations and expressions for its moments. Parameters are estimated using the maximum likelihood estimation method. A simulation study is carried out to observe the performance of the maximum likelihood estimators. Finally, we compare the efficiency of the new distribution with other distributions in the literature using a real data set. The study shows that the proposed approach presents satisfactory results.

scholarly journals Multivariate Skew-Power-Normal Distributions: Properties and Associated Inference

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1509
Author(s):  
Guillermo Martínez-Flórez ◽  
Artur J. Lemonte ◽  
Hugo S. Salinas
Keyword(s):  
Maximum Likelihood ◽  
Normal Distribution ◽  
Real Data ◽  
Likelihood Method ◽  
Model Parameters ◽  
Unknown Parameters ◽  
Multivariate Distributions ◽  
Likelihood Approach ◽  
Information Matrices ◽  
Univariate Distribution

The univariate power-normal distribution is quite useful for modeling many types of real data. On the other hand, multivariate extensions of this univariate distribution are not common in the statistic literature, mainly skewed multivariate extensions that can be bimodal, for example. In this paper, based on the univariate power-normal distribution, we extend the univariate power-normal distribution to the multivariate setup. Structural properties of the new multivariate distributions are established. We consider the maximum likelihood method to estimate the unknown parameters, and the observed and expected Fisher information matrices are also derived. Monte Carlo simulation results indicate that the maximum likelihood approach is quite effective to estimate the model parameters. An empirical application of the proposed multivariate distribution to real data is provided for illustrative purposes.

scholarly journals Working with Gaussian Random Noise for Multi-Sensor Archaeological Prospection: Fusion of Ground Penetrating Radar Depth Slices and Ground Spectral Signatures from 0.00 m to 0.60 m below Ground Surface

2019 ◽  
Vol 11 (16) ◽  
pp. 1895 ◽  
Author(s):  
Agapiou ◽  
Sarris
Keyword(s):  
Regression Model ◽  
Normal Distribution ◽  
Ground Penetrating Radar ◽  
Regression Models ◽  
Random Noise ◽  
Ground Surface ◽  
Archaeological Prospection ◽  
Spectral Signatures ◽  
Ground Penetrating ◽  
The Impact

The integration of different remote sensing datasets acquired from optical and radar sensors can improve the overall performance and detection rate for mapping sub-surface archaeological remains. However, data fusion remains a challenge for archaeological prospection studies, since remotely sensed sensors have different instrument principles, operating in different wavelengths. Recent studies have demonstrated that some fusion modelling can be achieved under ideal measurement conditions (e.g., simultaneously measurements in no hazy days) using advance regression models, like those of the nonlinear Bayesian Neural Networks. This paper aims to go a step further and investigate the impact of noise in regression models, between datasets obtained from ground-penetrating radar (GPR) and portable field spectroradiometers. Initially, the GPR measurements provided three depth slices of 20 cm thickness, starting from 0.00 m up to 0.60 m below the ground surface while ground spectral signatures acquired from the spectroradiometer were processed to calculate 13 multispectral and 53 hyperspectral indices. Then, various levels of Gaussian random noise ranging from 0.1 to 0.5 of a normal distribution, with mean 0 and variance 1, were added at both GPR and spectral signatures datasets. Afterward, Bayesian Neural Network regression fitting was applied between the radar (GPR) versus the optical (spectral signatures) datasets. Different regression model strategies were implemented and presented in the paper. The overall results show that fusion with a noise level of up to 0.2 of the normal distribution does not dramatically drop the regression model between the radar and optical datasets (compared to the non-noisy data). Finally, anomalies appearing as strong reflectors in the GPR measurements, continue to provide an obvious contrast even with noisy regression modelling.

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