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 New Lindley Half Cauchy Distribution: Theory and Applications

2020 ◽  
Vol 9 (4) ◽  
pp. 1-7
Keyword(s):  
Maximum Likelihood ◽  
Real Data ◽  
Cauchy Distribution ◽  
Least Square ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Estimation Methods ◽  
Model Parameters ◽  
Data Set ◽  
Von Mises

In this paper, we have defined a new two-parameter new Lindley half Cauchy (NLHC) distribution using Lindley-G family of distribution which accommodates increasing, decreasing and a variety of monotone failure rates. The statistical properties of the proposed distribution such as probability density function, cumulative distribution function, quantile, the measure of skewness and kurtosis are presented. We have briefly described the three well-known estimation methods namely maximum likelihood estimators (MLE), least-square (LSE) and Cramer-Von-Mises (CVM) methods. All the computations are performed in R software. By using the maximum likelihood method, we have constructed the asymptotic confidence interval for the model parameters. We verify empirically the potentiality of the new distribution in modeling a real data set.

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 A New Flexible Three-Parameter Model: Properties, Clayton Copula, and Modeling Real Data

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 440 ◽  
Author(s):  
Abdulhakim A. Al-babtain ◽  
I. Elbatal ◽  
Haitham M. Yousof
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Method ◽  
Real Data ◽  
Likelihood Method ◽  
Model Parameters ◽  
Simple Type ◽  
Data Sets ◽  
New Model ◽  
Clayton Copula ◽  
Mathematical Properties

In this article, we introduced a new extension of the binomial-exponential 2 distribution. We discussed some of its structural mathematical properties. A simple type Copula-based construction is also presented to construct the bivariate- and multivariate-type distributions. We estimated the model parameters via the maximum likelihood method. Finally, we illustrated the importance of the new model by the study of two real data applications to show the flexibility and potentiality of the new model in modeling skewed and symmetric data sets.

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 Lehmann Type II Frechet Poisson Distribution: Properties, Inference and Applications as a Life Time Distribution

2021 ◽  
Vol 10 (3) ◽  
pp. 8
Author(s):  
Adebisi Ade Ogunde ◽  
Gbenga Adelekan Olalude ◽  
Oyebimpe Emmanuel Adeniji ◽  
Kayode Balogun
Keyword(s):  
Maximum Likelihood ◽  
Poisson Distribution ◽  
Maximum Likelihood Method ◽  
Life Time ◽  
Real Data ◽  
Likelihood Method ◽  
Strength Parameter ◽  
Model Parameters ◽  
Type Ii ◽  
Time Data

A new generalization of the Frechet distribution called Lehmann Type II Frechet Poisson distribution is defined and studied. Various structural mathematical properties of the proposed model including ordinary moments, incomplete moments, generating functions, order statistics, Renyi entropy, stochastic ordering, Bonferroni and Lorenz curve, mean and median deviation, stress-strength parameter are investigated. The maximum likelihood method is used to estimate the model parameters. We examine the performance of the maximum likelihood method by means of a numerical simulation study. The new distribution is applied for modeling three real data sets to illustrate empirically its flexibility and tractability in modeling life time data.

scholarly journals An algorithm for the assessment of several small rates

2012 ◽  
Vol 53 ◽  
Author(s):  
Leonidas Sakalauskas ◽  
Ingrida Vaičiulytė
Keyword(s):  
Maximum Likelihood ◽  
Bayesian Approach ◽  
Maximum Likelihood Method ◽  
Rare Events ◽  
Maximum Likelihood Estimates ◽  
Likelihood Method ◽  
Model Parameters ◽  
Unknown Parameters ◽  
Empirical Bayesian ◽  
Empirical Bayesian Approach

The present paper describes the empirical Bayesian approach applied in the estimation of several small rates. Modeling by empirical Bayesian approach the probabilities of several rare events, it is assumed that the frequencies of events follow to Poisson’s law with different parameters, which are correlated Gaussian random values. The unknown parameters are estimated by the maximum likelihood method computing the integrals appeared here by Hermite–Gauss quadratures. The equations derived that are satisfied by maximum likelihood estimates of model parameters.

scholarly journals Generalized Exponential Power Distribution with Application to Complete and Censored Data

2021 ◽  
pp. 34-55
Author(s):  
M. M. E. Abd El-Monsef ◽  
M. M. El-Awady
Keyword(s):  
Maximum Likelihood ◽  
Censored Data ◽  
Power Distribution ◽  
Real Data ◽  
Likelihood Method ◽  
Model Parameters ◽  
Type I ◽  
Sample Type ◽  
Exponential Power Distribution ◽  
Exponential Power

The exponential power distribution (EP) is a lifetime model that can exhibit increasing and bathtub hazard rate function. This paper proposed a generalization of EP distribution, named generalized exponential power (GEP) distribution. Some properties of GEP distribution will be investigated. Recurrence relations for single moments of generalized ordered statistics from GEP distribution are established and used for characterizing the GEP distribution. Estimation of the model parameters are derived using maximum likelihood method based on complete sample, type I, type II and random censored samples. A simulation study is performed in order to examine the accuracy of the maximum likelihood estimators of the model parameters. Three applications to real data, two with censored data, are provided in order to show the superiority of the proposed model to other models.

scholarly journals Half-Logistic Xgamma Distribution: Properties and Estimation under Censored Samples

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Rashad Bantan ◽  
Amal S. Hassan ◽  
Mahmoud Elsehetry ◽  
B. M. Golam Kibria
Keyword(s):  
Maximum Likelihood ◽  
Residual Life ◽  
Stochastic Ordering ◽  
Real Data ◽  
Least Square ◽  
Likelihood Method ◽  
Mean Residual Life ◽  
Model Parameters ◽  
Censored Samples ◽  
Von Mises

This paper proposed a new probability distribution, namely, the half-logistic xgamma (HLXG) distribution. Various statistical properties, such as, moments, incomplete moments, mean residual life, and stochastic ordering of the proposed distribution, are discussed. Parameter estimation of the half-logistic xgamma distribution is approached by the maximum likelihood method based on complete and censored samples. Asymptotic confidence intervals of model parameters are provided. A simulation study is conducted to illustrate the theoretical results. Moreover, the model parameters of the HLXG distribution are estimated by using the maximum likelihood, least square, maximum product spacing, percentile, and Cramer–von Mises (CVM) methods. Superiority of the new model over some existing distributions is illustrated through three real data sets.

scholarly journals A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1439
Author(s):  
Guillermo Martínez-Flórez ◽  
Víctor Leiva ◽  
Emilio Gómez-Déniz ◽  
Carolina Marchant
Keyword(s):  
Normal Distribution ◽  
Real Data ◽  
Continuous Variable ◽  
Likelihood Method ◽  
Skew Normal Distribution ◽  
Normal Distributions ◽  
New Type ◽  
Skew Normal Distributions ◽  
Information Matrices ◽  
Skew Normal

In this paper, we consider skew-normal distributions for constructing new a distribution which allows us to model proportions and rates with zero/one inflation as an alternative to the inflated beta distributions. The new distribution is a mixture between a Bernoulli distribution for explaining the zero/one excess and a censored skew-normal distribution for the continuous variable. The maximum likelihood method is used for parameter estimation. Observed and expected Fisher information matrices are derived to conduct likelihood-based inference in this new type skew-normal distribution. Given the flexibility of the new distributions, we are able to show, in real data scenarios, the good performance of our proposal.

scholarly journals A New Parametric Life Family of Distributions: Properties, Copula and Modeling Failure and Service Times

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1462
Author(s):  
Mansour Shrahili ◽  
Naif Alotaibi
Keyword(s):  
Maximum Likelihood ◽  
Probability Distributions ◽  
Real Life ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Real Data ◽  
Likelihood Method ◽  
Model Parameters ◽  
New Family ◽  
Renyi’S Entropy

A new family of probability distributions is defined and applied for modeling symmetric real-life datasets. Some new bivariate type G families using Farlie–Gumbel–Morgenstern copula, modified Farlie–Gumbel–Morgenstern copula, Clayton copula and Renyi’s entropy copula are derived. Moreover, some of its statistical properties are presented and studied. Next, the maximum likelihood estimation method is used. A graphical assessment based on biases and mean squared errors is introduced. Based on this assessment, the maximum likelihood method performs well and can be used for estimating the model parameters. Finally, two symmetric real-life applications to illustrate the importance and flexibility of the new family are proposed. The symmetricity of the real data is proved nonparametrically using the kernel density estimation method.

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