The Theory And Application Of Skew Symmetric Normal Distribution

The distribution of data generated in practical activities such as engineering sciences,biomedicine,social sciences,climate science,financial economics,and social production is not necessarily strictly symmetrical,it may be skew distribution.If this kind of data is fitted with a symmetrical probability statistical model like normal distribution,the distribution characteristics of data can not be described correctly.However,the distribution characteristics of these data can be described more accurately by skew distribution fitting.Therefore,the study of skew distribution is of great significance.Since Azzalini proposed the skew normal distribution,more and more scholars began to study the skew distribution,and successively proposed the skew-symmetric-distribution,skew-normal-t-distribution,?-skew-normal-distribution models,etc.,and studied their properties,providing a theoretical basis for further study of their applications in different types of skewness distribution data.Based on this consideration,this paper defines a new kind of skewness distribution-skew symmetric normal distribution.This paper provides a new fitting model for the data under the skewed distribution,and studies its related properties and its application in regression,including parameter estimation,significance test of regression coefficient,homogeneity test of scale parameters and skewness parameters,etc.The main research contents include the following four aspectsFirstly,the skew-symmetric-normal-distribution is proposed,and its related properties is studied.Some skew-symmetric-normal-distributions are enumerated,including a skew normal distribution,skew Logistic normal distribution and skew uniform normal distribution,skew t normal distribution,skew Laplace normal distribution,skew triangle normal distribution,etc The numerical characteristics and some properties of them are discussed,and the process of solving the maximum likelihood estimation of skew symmetric normal distribution parameters by using the general and improved gauss-newton iterative method and the related expressions are derived,which are verified by Monte Carlo numerical simulation analysisSecondly,an improved gauss-newton iterative method for maximum likelihood estimation of the parameters of a skew normal multiple linear regression model is introduced This paper studies the application of likelihood ratio test and Score test in the significance test of coefficient,the homogeneity test of skewness parameter and scale parameter of skew normal multivariate linear regression model,deduces the analytical expression of correlation test statistic,and explains it by Monte Carlo numerical simulationThirdly,the method of Maximum likelihood estimation for parameters of skew Logistic normal multiple linear regression model is given,and the analytical expression of the second skew derivative matrix of its log-likelihood function is derived.This paper studies the application of likelihood ratio test and Score test in the significance test of coefficient of skew Logistic normal multivariate linear regression model,the homogeneity test of skewness parameter and scale parameter,deduces the analytical expression of correlation test statistic,and verifies it by Monte Carlo numerical simulation analysis.Fourthly,the case data of housing price in Boston are fitted with the multivariate linear regression models of normal,skew normal and skew Logistic normal.The paper introduces the method of solving maximum likelihood estimation of model parameters,significance test of regression coefficient,homogeneity test of skewness parameter and scale parameter on example data.The Akaike information criterion(AIC)and Bayesian information criterion(BIC)were used to evaluate the applicability of different regression models on instance data.