A Power Maxwell Distribution with Heavy Tails and Applications

In this paper we introduce a distribution which is an extension of the power Maxwell distribution. This new distribution is constructed based on the quotient of two independent random variables, the distributions of which are the power Maxwell distribution and a function of the uniform distribution (0,1) respectively. Thus the result is a distribution with greater kurtosis than the power Maxwell. We study the general density of this distribution, and some properties, moments, asymmetry and kurtosis coefficients. Maximum likelihood and moments estimators are studied. We also develop the expectation–maximization algorithm to make a simulation study and present two applications to real data.

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An Asymmetric Distribution with Heavy Tails and Its Expectation–Maximization (EM) Algorithm Implementation

Symmetry ◽

10.3390/sym11091150 ◽

2019 ◽

Vol 11 (9) ◽

pp. 1150 ◽

Cited By ~ 1

Author(s):

Neveka M. Olmos ◽

Osvaldo Venegas ◽

Yolanda M. Gómez ◽

Yuri A. Iriarte

Keyword(s):

Expectation Maximization ◽

Heavy Tails ◽

Expectation Maximization Algorithm ◽

Real Data ◽

Independent Random Variables ◽

Asymmetric Distribution ◽

Data Sets ◽

Normal Distributions ◽

The Moment ◽

Algorithm Implementation

In this paper we introduce a new distribution constructed on the basis of the quotient of two independent random variables whose distributions are the half-normal distribution and a power of the exponential distribution with parameter 2 respectively. The result is a distribution with greater kurtosis than the well known half-normal and slashed half-normal distributions. We studied the general density function of this distribution, with some of its properties, moments, and its coefficients of asymmetry and kurtosis. We developed the expectation–maximization algorithm and present a simulation study. We calculated the moment and maximum likelihood estimators and present three illustrations in real data sets to show the flexibility of the new model.

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Slashed Exponentiated Rayleigh Distribution

Revista Colombiana de Estadística ◽

10.15446/rce.v38n2.51673 ◽

2015 ◽

Vol 38 (2) ◽

pp. 453-466 ◽

Cited By ~ 1

Author(s):

Hugo S. Salinas ◽

Yuri A. Iriarte ◽

Heleno Bolfarine

Keyword(s):

Maximum Likelihood ◽

Maximum Likelihood Method ◽

Real Data ◽

Random Variable ◽

Rayleigh Distribution ◽

Likelihood Method ◽

Independent Random Variables ◽

Parameter Estimates ◽

Positive Data ◽

New Distribution

<p>In this paper we introduce a new distribution for modeling positive data with high kurtosis. This distribution can be seen as an extension of the exponentiated Rayleigh distribution. This extension builds on the quotient of two independent random variables, one exponentiated Rayleigh in the numerator and Beta(q,1) in the denominator with q>0. It is called the slashed exponentiated Rayleigh random variable. There is evidence that the distribution of this new variable can be more flexible in terms of modeling the kurtosis regarding the exponentiated Rayleigh distribution. The properties of this distribution are studied and the parameter estimates are calculated using the maximum likelihood method. An application with real data reveals good performance of this new distribution.</p>

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A Cognitive Diagnosis Model for Identifying Coexisting Skills and Misconceptions

Applied Psychological Measurement ◽

10.1177/0146621617722791 ◽

2017 ◽

Vol 42 (3) ◽

pp. 179-191 ◽

Cited By ~ 7

Author(s):

Bor-Chen Kuo ◽

Chun-Hua Chen ◽

Jimmy de la Torre

Keyword(s):

Simulation Study ◽

Expectation Maximization ◽

Expectation Maximization Algorithm ◽

Real Data ◽

Cognitive Diagnosis ◽

Future Research ◽

Model Parameters ◽

Cognitive Diagnosis Models ◽

Proposed Model ◽

Diagnosis Model

At present, most existing cognitive diagnosis models (CDMs) are designed to either identify the presence and absence of skills or misconceptions, but not both. This article proposes a CDM that can be used to simultaneously identify what skills and misconceptions students possess. In addition, it proposes the use of the expectation-maximization algorithm to estimate the model parameters. A simulation study is conducted to evaluate the viability of the proposed model and algorithm. Real data are analyzed to demonstrate the applicability of the proposed model, and compare it with existing CDMs. Furthermore, a real data–based simulation study is conducted to determine how the correct classification rates in the context of the proposed model can be improved. Issues related to the proposed model and future research are discussed.

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Robust mixture regression based on the skew t distribution

Revista Colombiana de Estadística ◽

10.15446/rce.v40n1.53580 ◽

2017 ◽

Vol 40 (1) ◽

pp. 45-64 ◽

Cited By ~ 7

Author(s):

Fatma Zehra Doğru ◽

Olcay Arslan

Keyword(s):

Maximum Likelihood ◽

Em Algorithm ◽

Simulation Study ◽

Expectation Maximization ◽

Real Data ◽

Scale Mixture ◽

Mixture Regression ◽

Regression Procedure ◽

Heavy Tailed ◽

T Distribution

In this study, we propose a robust mixture regression procedure based on the skew t distribution to model heavy-tailed and/or skewed errors in a mixture regression setting. Using the scale mixture representation of the skew t distribution, we give an Expectation Maximization (EM) algorithm to compute the maximum likelihood (ML) estimates for the paramaters of interest. The performance of proposed estimators is demonstrated by a simulation study and a real data example.

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Scale Mixture of Rayleigh Distribution

Mathematics ◽

10.3390/math8101842 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1842

Author(s):

Pilar A. Rivera ◽

Inmaculada Barranco-Chamorro ◽

Diego I. Gallardo ◽

Héctor W. Gómez

Keyword(s):

Maximum Likelihood ◽

Em Algorithm ◽

Gamma Distribution ◽

Simulation Study ◽

Expectation Maximization ◽

Rayleigh Distribution ◽

Independent Random Variables ◽

Scale Mixture ◽

New Model ◽

Lifetime Analysis

In this paper, the scale mixture of Rayleigh (SMR) distribution is introduced. It is proven that this new model, initially defined as the quotient of two independent random variables, can be expressed as a scale mixture of a Rayleigh and a particular Generalized Gamma distribution. Closed expressions are obtained for its pdf, cdf, moments, asymmetry and kurtosis coefficients. Its lifetime analysis, properties and Rényi entropy are studied. Inference based on moments and maximum likelihood (ML) is proposed. An Expectation-Maximization (EM) algorithm is implemented to estimate the parameters via ML. This algorithm is also used in a simulation study, which illustrates the good performance of our proposal. Two real datasets are considered in which it is shown that the SMR model provides a good fit and it is more flexible, especially as for kurtosis, than other competitor models, such as the slashed Rayleigh distribution.

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Transforming variables to central normality

Machine Learning ◽

10.1007/s10994-021-05960-5 ◽

2021 ◽

Author(s):

Jakob Raymaekers ◽

Peter J. Rousseeuw

Keyword(s):

Maximum Likelihood ◽

Maximum Likelihood Estimator ◽

Simulation Study ◽

Real Data ◽

Data Sets ◽

Transformation Parameter ◽

Likelihood Estimator ◽

Extensive Simulation ◽

Highly Sensitive

AbstractMany real data sets contain numerical features (variables) whose distribution is far from normal (Gaussian). Instead, their distribution is often skewed. In order to handle such data it is customary to preprocess the variables to make them more normal. The Box–Cox and Yeo–Johnson transformations are well-known tools for this. However, the standard maximum likelihood estimator of their transformation parameter is highly sensitive to outliers, and will often try to move outliers inward at the expense of the normality of the central part of the data. We propose a modification of these transformations as well as an estimator of the transformation parameter that is robust to outliers, so the transformed data can be approximately normal in the center and a few outliers may deviate from it. It compares favorably to existing techniques in an extensive simulation study and on real data.

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