Alpha Skew T Distribution

The skew distribution usually refers to the addition of one or more skewing parame?ters based on the symmetric distribution,so that the new distribution includes not only the symmetric case but,also the asymmetric(ie,skew)case.Because of the processing methods on symmetric distribution are flexible,and the data appearing in practical applications are often asymmetrical,therefore,the skew distribution is attracting more and more attention of scholars.The research on skew symmetry distribution has been s-tarted as early as 1985.Azzalini first proposed the skew normal distribution,studied the basic properties of skew normal distribution,and generalized to the high-dimensional situation.Subsequently,the t distribution,Laplace distribution,Cauchy distribution,Slash distribution,Logistic distribution,etc.,received extensively research,and applied to actual situations.And these symmetrical distribution of skew methods are basically follow the method proposed by Azzalini.Follow-up studies are mostly concentrated in the properties and applications of skew symmetric distributions,while new methods of skewness were seldom researched.This paper proposes the ? skew t distribution based on the ? skew normal distribution,and studies the statistical properties of this distribut,ion and its parameter estimation problem.The first chapter introduces the research background of the skew symmetry distri-bution,lists the existing skew t distributions and the relationship among these distribu-tions,gives some main theorems used in this paper.Chapter 2 gives the basic definition of the ? skew t distribution,and discusses its density function,distribution function,eigenfunction,limit properties,moments,and random number generation.Chapters 3 and 4 are the point estimation and its properties of the distribution,specifically the existence and consistency of the moment estimation in the case of three unknown parameters,the asymptotic normality of the moment estimation in the case of two unknown parameters,as well as the existence,consistency and asymptotic normality of maximum likelihood estimates under large samples.Chapter 5 gives the interval estimation of the distribution,including single-parameter interval estimation and joint interval estimation.In Chapter 6,the maximum likelihood estimation of the a skew t distribution is performed using the Mento-Carlo simulation of point estimation and interval estima-tion.The effect of the estimation under different sample sizes and the coverage rate of asymptotic confidence interval are investigated.The seventh chapter is the performance of the distribution on several sets of real data,including a set of unimodal data and two sets of bimodal data,and compares the fitting effect of the distribution with the existing several distributions.