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 New Flexible Regression Models Generated by Gamma Random Variables with Censored Data

2016 ◽  
Vol 5 (3) ◽  
pp. 9 ◽  
Author(s):  
Elizabeth M. Hashimoto ◽  
Gauss M. Cordeiro ◽  
Edwin M.M. Ortega ◽  
G.G. Hamedani
Keyword(s):  
Regression Model ◽  
Censored Data ◽  
Survival Data ◽  
Regression Models ◽  
Empirical Distribution ◽  
Real Data ◽  
Maximum Likelihood Estimates ◽  
Standard Normal Distribution ◽  
Model Parameters ◽  
Linear Regression Models

We propose and study a new log-gamma Weibull regression model. We obtain explicit expressions for the raw and incomplete moments, quantile and generating functions and mean deviations of the log-gamma Weibull distribution. We demonstrate that the new regression model can be applied to censored data since it represents a parametric family of models which includes as sub-models several widely-known regression models and therefore can be used more effectively in the analysis of survival data. We obtain the maximum likelihood estimates of the model parameters by considering censored data and evaluate local influence on the estimates of the parameters by taking different perturbation schemes. Some global-influence measurements are also investigated. Further, for different parameter settings, sample sizes and censoring percentages, various simulations are performed. In addition, the empirical distribution of some modified residuals are displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the proposed regression model applied to censored data. We demonstrate that our extended regression model is very useful to the analysis of real data and may give more realistic fits than other special regression models. 

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 A New Log-location Regression Model with Influence Diagnostics and Residual Analysis

2018 ◽  
pp. 417
Author(s):  
Emrah Altun ◽  
Haitham M. Yousof ◽  
GG Hamedani
Keyword(s):  
Regression Model ◽  
Estimation Method ◽  
Real Data ◽  
Distance Measures ◽  
Likelihood Method ◽  
Model Parameters ◽  
Unknown Parameters ◽  
Data Set ◽  
Cook Distance ◽  
Influence Diagnostics

A new four-parameter lifetime model called OddLog-Logistic Burr XII distribution, is defined and investigated. Some of itsmathematical properties are derived. Some useful characterization resultsbased on \ the ratio of two truncated moments, based on the hazard functionas well as on the conditional expectation of certain functions of the randomvariable are presented. The maximum likelihood method is used to estimatethe model parameters by means of a graphical Monte Carlo simulation study.Moreover, we introduce a new log-location regression model based on theproposed distribution. The Jackknife estimation method as an alternativemethod is used to estimate the unknown parameters of new regression model. Thegeneralized cook distance and likelihood distance measures are used todetect the possible influential observations. The martingale and modifieddeviance residuals are defined to detect outliers and evaluate the modelassumptions. The potentiality of the new regression model is illustrated bymeans of a real data set.

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.

scholarly journals Theory and Applications of the Unit Gamma/Gompertz Distribution

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1850
Author(s):  
Rashad A. R. Bantan ◽  
Farrukh Jamal ◽  
Christophe Chesneau ◽  
Mohammed Elgarhy
Keyword(s):  
Stochastic Ordering ◽  
Real Data ◽  
Rate Function ◽  
The Other ◽  
Likelihood Method ◽  
Model Parameters ◽  
Data Sets ◽  
Gompertz Distribution ◽  
Probability And Statistics ◽  
Analytical Behavior

Unit distributions are commonly used in probability and statistics to describe useful quantities with values between 0 and 1, such as proportions, probabilities, and percentages. Some unit distributions are defined in a natural analytical manner, and the others are derived through the transformation of an existing distribution defined in a greater domain. In this article, we introduce the unit gamma/Gompertz distribution, founded on the inverse-exponential scheme and the gamma/Gompertz distribution. The gamma/Gompertz distribution is known to be a very flexible three-parameter lifetime distribution, and we aim to transpose this flexibility to the unit interval. First, we check this aspect with the analytical behavior of the primary functions. It is shown that the probability density function can be increasing, decreasing, “increasing-decreasing” and “decreasing-increasing”, with pliant asymmetric properties. On the other hand, the hazard rate function has monotonically increasing, decreasing, or constant shapes. We complete the theoretical part with some propositions on stochastic ordering, moments, quantiles, and the reliability coefficient. Practically, to estimate the model parameters from unit data, the maximum likelihood method is used. We present some simulation results to evaluate this method. Two applications using real data sets, one on trade shares and the other on flood levels, demonstrate the importance of the new model when compared to other unit models.

scholarly journals Mixture of Inverse Weibull and Lognormal Distributions: Properties, Estimation, and Illustration

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
K. S. Sultan ◽  
A. S. Al-Moisheer
Keyword(s):  
Information Matrix ◽  
Real Data ◽  
Likelihood Method ◽  
Model Parameters ◽  
Component Mixture ◽  
Data Set ◽  
Hazard Functions ◽  
Lognormal Distributions ◽  
Proposed Model ◽  
Estimation Of Model Parameters

We discuss the two-component mixture of the inverse Weibull and lognormal distributions (MIWLND) as a lifetime model. First, we discuss the properties of the proposed model including the reliability and hazard functions. Next, we discuss the estimation of model parameters by using the maximum likelihood method (MLEs). We also derive expressions for the elements of the Fisher information matrix. Next, we demonstrate the usefulness of the proposed model by fitting it to a real data set. Finally, we draw some concluding remarks.

scholarly journals A New Linear Regression Kalman Filter with Symmetric Samples

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2139
Author(s):  
Xiuqiong Chen ◽  
Jiayi Kang ◽  
Mina Teicher ◽  
Stephen S.-T. Yau
Keyword(s):  
Kalman Filter ◽  
Linear Regression ◽  
Particle Filter ◽  
Normal Distribution ◽  
Extended Kalman Filter ◽  
Nonlinear Filtering ◽  
Unscented Kalman Filter ◽  
Standard Normal Distribution ◽  
Leibler Divergence ◽  
Standard Normal

Nonlinear filtering is of great significance in industries. In this work, we develop a new linear regression Kalman filter for discrete nonlinear filtering problems. Under the framework of linear regression Kalman filter, the key step is minimizing the Kullback–Leibler divergence between standard normal distribution and its Dirac mixture approximation formed by symmetric samples so that we can obtain a set of samples which can capture the information of reference density. The samples representing the conditional densities evolve in a deterministic way, and therefore we need less samples compared with particle filter, as there is less variance in our method. The numerical results show that the new algorithm is more efficient compared with the widely used extended Kalman filter, unscented Kalman filter and particle filter.

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