scholarly journals A New Extended Mixture Skew Normal Distribution, With Applications

2019 ◽  
Vol 42 (2) ◽  
pp. 167-183
Author(s):  
Haroon M. Barakat ◽  
Abdallh W. Aboutahoun ◽  
Naeema El-kadar
Keyword(s):  
Normal Distribution ◽  
Location Parameter ◽  
Distribution Functions ◽  
Maximum Likelihood Estimates ◽  
Computational Techniques ◽  
Model Parameters ◽  
Skew Normal Distribution ◽  
Data Set ◽  
Wide Range ◽  
Skew Normal

One of the most important property of the mixture normal distributions-model is its flexibility to accommodate various types of distribution functions (df's). We show that the mixture of the skew normal distribution and its reverse, after adding a location parameter to the skew normal distribution, and adding the same location parameter with different sign to its reverse is a family of df's that contains all the possible types of df's. Besides, it has a very remarkable wide range of the indices of skewness and kurtosis. Computational techniques using EM-type algorithms are employed for iteratively computing maximum likelihood estimates of the model parameters. Moreover, an application with a body mass index real data set is presented.

scholarly journals Hypothesis Testing on the Location Parameter of a Skew-Normal Distribution (SND) with Application

2018 ◽  
Vol 20 ◽  
pp. 03003
Author(s):  
Phontita Thiuthad ◽  
Nabendu Pal
Keyword(s):  
Normal Distribution ◽  
Real Life ◽  
Location Parameter ◽  
Test Methods ◽  
Skew Normal Distribution ◽  
Data Set ◽  
Sample Mean ◽  
Real Life Data ◽  
Approximate Probability ◽  
Skew Normal

This work deals with testing a hypothesis on the location parameter (μ) of a skew-normal distribution (SND) based on a random sample of size n. The details of this work can be summarized in four major components: (a) First we review some useful results on SND, including the approximate probability distribution of the sample average. (b) Next, we develop several tests to test a hypothesis on μ based on the sample mean when the scale (σ) and shape (λ) parameters are known. (c) The tests for the known scale and shape are then extended for unknown scale and shape. (d) Finally, the test methods have been used for a real-life data set.

scholarly journals Properties and Applications of a New Family of Skew Distributions

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 87
Author(s):  
Emilio Gómez-Déniz ◽  
Barry C. Arnold ◽  
José M. Sarabia ◽  
Héctor W. Gómez
Keyword(s):  
Normal Distribution ◽  
Continuous Distribution ◽  
Central Moment ◽  
Distribution Functions ◽  
Logistic Distribution ◽  
Skew Normal Distribution ◽  
Data Set ◽  
Closed Expression ◽  
Parent Distribution ◽  
Skew Normal

We introduce two families of continuous distribution functions with not-necessarily symmetric densities, which contain a parent distribution as a special case. The two families proposed depend on two parameters and are presented as an alternative to the skew normal distribution and other proposals in the statistical literature. The density functions of these new families are given by a closed expression which allows us to easily compute probabilities, moments and related quantities. The second family can exhibit bimodality and its standardized fourth central moment (kurtosis) can be lower than that of the Azzalini skew normal distribution. Since the second proposed family can be bimodal we fit two well-known data set with this feature as applications. We concentrate attention on the case in which the normal distribution is the parent distribution but some consideration is given to other parent distributions, such as the logistic distribution.

scholarly journals Improved Approximation of the Sum of Random Vectors by the Skew Normal Distribution

2014 ◽  
Vol 51 (2) ◽  
pp. 466-482 ◽  
Author(s):  
Marcus C. Christiansen ◽  
Nicola Loperfido
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Distribution Functions ◽  
Skew Normal Distribution ◽  
Random Vectors ◽  
Multivariate Distributions ◽  
Uniform Distance ◽  
Special Case ◽  
Independent Identically Distributed ◽  
Skew Normal

We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n-2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.

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 Improved Approximation of the Sum of Random Vectors by the Skew Normal Distribution

2014 ◽  
Vol 51 (02) ◽  
pp. 466-482 ◽  
Author(s):  
Marcus C. Christiansen ◽  
Nicola Loperfido
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Distribution Functions ◽  
Skew Normal Distribution ◽  
Random Vectors ◽  
Multivariate Distributions ◽  
Uniform Distance ◽  
Special Case ◽  
Independent Identically Distributed ◽  
Skew Normal

We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n -2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.

scholarly journals Approximating the Distribution of the Product of Two Normally Distributed Random Variables

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1201 ◽  
Author(s):  
Antonio Seijas-Macías ◽  
Amílcar Oliveira ◽  
Teresa A. Oliveira ◽  
Víctor Leiva
Keyword(s):  
Normal Distribution ◽  
Random Variables ◽  
Graphical Analysis ◽  
Early Years ◽  
Computational Techniques ◽  
Skew Normal Distribution ◽  
Coefficients Of Variation ◽  
Better Than ◽  
Skew Normal ◽  
Normally Distributed

The distribution of the product of two normally distributed random variables has been an open problem from the early years in the XXth century. First approaches tried to determinate the mathematical and statistical properties of the distribution of such a product using different types of functions. Recently, an improvement in computational techniques has performed new approaches for calculating related integrals by using numerical integration. Another approach is to adopt any other distribution to approximate the probability density function of this product. The skew-normal distribution is a generalization of the normal distribution which considers skewness making it flexible. In this work, we approximate the distribution of the product of two normally distributed random variables using a type of skew-normal distribution. The influence of the parameters of the two normal distributions on the approximation is explored. When one of the normally distributed variables has an inverse coefficient of variation greater than one, our approximation performs better than when both normally distributed variables have inverse coefficients of variation less than one. A graphical analysis visually shows the superiority of our approach in relation to other approaches proposed in the literature on the topic.

scholarly journals Bayesian Inference of Finite Population Quantiles for Skewed Survey Data Using Skew-Normal Penalized Spline Regression

2019 ◽  
Vol 8 (4) ◽  
pp. 792-816
Author(s):  
Yutao Liu ◽  
Qixuan Chen
Keyword(s):  
Normal Distribution ◽  
Survey Data ◽  
Finite Population ◽  
Location Parameter ◽  
Penalized Splines ◽  
Sample Survey ◽  
Skew Normal Distribution ◽  
Scale Parameters ◽  
Penalized Spline ◽  
Skew Normal

Abstract Skewed data are common in sample surveys. In probability proportional to size sampling, we propose two Bayesian model-based predictive methods for estimating finite population quantiles with skewed sample survey data. We assume the survey outcome to follow a skew-normal distribution given the probability of selection and model the location and scale parameters of the skew-normal distribution as functions of the probability of selection. To allow a flexible association between the survey outcome and the probability of selection, the first method models the location parameter with a penalized spline and the scale parameter with a polynomial function, while the second method models both the location and scale parameters with penalized splines. Using a fully Bayesian approach, we obtain the posterior predictive distributions of the nonsampled units in the population and thus the posterior distributions of the finite population quantiles. We show through simulations that our proposed methods are more efficient and yield shorter credible intervals with better coverage rates than the conventional weighted method in estimating finite population quantiles. We demonstrate the application of our proposed methods using data from the 2013 National Drug Abuse Treatment System Survey.

scholarly journals Skewed and Flexible Skewed Distributions: A Modern Look at the Distribution of BMI

2017 ◽  
Vol 14 (2) ◽  
Author(s):  
Thao Tran ◽  
Cara Wiskow ◽  
Mohammad Aziz
Keyword(s):  
Body Mass Index ◽  
Normal Distribution ◽  
Body Mass ◽  
Model Parameters ◽  
Skew Normal Distribution ◽  
Scale Mixture ◽  
Skewed Distributions ◽  
Goodness Of Fit Test ◽  
T Distribution ◽  
Skew Normal

The purpose of this study is to find distributions that best model body mass index (BMI) data. BMI has become a standard health indicator and numerous studies have been done to examine the distribution of BMI. Due to the skew and bimodal nature, we focus on modeling BMI with flexible skewed distributions. The distributions are fitted to University of Wisconsin–Eau Claire (UWEC) BMI data and to a data obtained from National Health and Nutrition Survey (NHANES). The model parameters are obtained using maximum likelihood estimation method. We compare flexible models to more conventional distributions, such as skew-normal, and skew-t distributions using AIC and BIC and Kolmogorov-Smirnov (K-S) goodness-of-fit test. Our results indicate that the skew-t and Alpha-Skew-Laplace distributions are reasonably competitive when describing unimodal BMI data whereas Alpha-Skew-Laplace and finite mixture of scale mixture of skew-normal and skew-t distributions are better alternatives to both unimodal and bimodal conventional distributions. The results we obtained are useful because we believe the models discussed in ours study will offer a framework for testing features such as bimodality, asymmetry, and robustness of the BMI data, thus providing a more detailed and accurate understanding of the distribution of BMI. KEYWORDS: Body Mass Index; Skew-normal distribution; Skew-t distribution; Flexible skewed distributions; Mixture distributions; Scale mixture of skew-normal distribution; K-S test

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