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

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.

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

Journal of Applied Probability ◽

10.1239/jap/1402578637 ◽

2014 ◽

Vol 51 (2) ◽

pp. 466-482 ◽

Cited By ~ 8

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.

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A New Extended Mixture Skew Normal Distribution, With Applications

Revista Colombiana de Estadística ◽

10.15446/rce.v42n2.70087 ◽

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.

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The Skew Normal Distribution

10.1093/oso/9780198505044.003.0012 ◽

2017 ◽

Author(s):

Russell Cheng

Keyword(s):

Normal Distribution ◽

Information Matrix ◽

Standard Theory ◽

Parameter Estimates ◽

Skew Normal Distribution ◽

True Parameter ◽

Folded Normal Distribution ◽

Natural Parametrization ◽

Special Case ◽

Skew Normal

This chapter considers the univariate skew-normal distribution, a generalization of the normal that includes the normal as a special case. The most natural parametrization is non-standard. This is because the Fisher information matrix is then singular at the true parameter value when the true model is the normal special case. The log-likelihood is then particularly flat in a certain coordinate direction. Standard theory cannot then be used to calculate the asymptotic distribution of all the parameter estimates. This problem can be handled using an alternative parametrization. There is another special case: the half/folded normal distribution. This occurs in the usual parametrization when the shape parameter is infinite. This is not a problem computationally and is easily handled. There are many generalizations to skew-t distributions and to tractable multivariate forms and regression versions. A very brief review is included of these.

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Properties and Applications of a New Family of Skew Distributions

Mathematics ◽

10.3390/math9010087 ◽

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.

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EM algorithm using overparameterization for the multivariate skew-normal distribution

Econometrics and Statistics ◽

10.1016/j.ecosta.2021.03.003 ◽

2021 ◽

Author(s):

Toshihiro Abe ◽

Hironori Fujisawa ◽

Takayuki Kawashima ◽

Christophe Ley

Keyword(s):

Em Algorithm ◽

Normal Distribution ◽

Skew Normal Distribution ◽

Skew Normal

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Parameter estimation for univariate Skew-Normal distribution based on the modified empirical characteristic function

Communication in Statistics- Theory and Methods ◽

10.1080/03610926.2021.1883655 ◽

2021 ◽

pp. 1-12

Author(s):

Gege Hou ◽

Ancha Xu ◽

Fengjing Cai ◽

You-Gan Wang

Keyword(s):

Parameter Estimation ◽

Characteristic Function ◽

Normal Distribution ◽

Empirical Characteristic Function ◽

Skew Normal Distribution ◽

Skew Normal

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Some properties of the unified skew-normal distribution

Statistical Papers ◽

10.1007/s00362-021-01235-2 ◽

2021 ◽

Author(s):

Reinaldo B. Arellano-Valle ◽

Adelchi Azzalini

Keyword(s):

Normal Distribution ◽

Fourth Order ◽

Probability Distributions ◽

Skew Normal Distribution ◽

Present Contribution ◽

The Family ◽

Multivariate Skewness ◽

Unified Skew Normal Distribution ◽

Skewness And Kurtosis ◽

Skew Normal

AbstractFor the family of multivariate probability distributions variously denoted as unified skew-normal, closed skew-normal and other names, a number of properties are already known, but many others are not, even some basic ones. The present contribution aims at filling some of the missing gaps. Specifically, the moments up to the fourth order are obtained, and from here the expressions of the Mardia’s measures of multivariate skewness and kurtosis. Other results concern the property of log-concavity of the distribution, closure with respect to conditioning on intervals, and a possible alternative parameterization.

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Copulaesque Versions of the Skew-Normal and Skew-Student Distributions

Symmetry ◽

10.3390/sym13050815 ◽

2021 ◽

Vol 13 (5) ◽

pp. 815

Author(s):

Christopher Adcock

Keyword(s):

Distribution Function ◽

Normal Distribution ◽

Degrees Of Freedom ◽

Special Functions ◽

Normal Distribution Function ◽

Skew Normal Distribution ◽

Marginal Distributions ◽

Student’S T ◽

Normal Copula ◽

Skew Normal

A recent paper presents an extension of the skew-normal distribution which is a copula. Under this model, the standardized marginal distributions are standard normal. The copula itself depends on the familiar skewing construction based on the normal distribution function. This paper is concerned with two topics. First, the paper presents a number of extensions of the skew-normal copula. Notably these include a case in which the standardized marginal distributions are Student’s t, with different degrees of freedom allowed for each margin. In this case the skewing function need not be the distribution function for Student’s t, but can depend on certain of the special functions. Secondly, several multivariate versions of the skew-normal copula model are presented. The paper contains several illustrative examples.

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