Research On The Asymptotic And Collapsibility Of Multidimensional T-Distribution

Since Gosset first proposed the t-distribution in 1908,many scholars and experts have continuously researched and expanded the properties of t-distribution based on it.Based on the characteristics and properties of t-distribution,it has been widely used in various fields.Further research on it Uniformly asymptotic normality and collapsibility with covariates,which makes it easier to use t-distribution to do probability integration in practical problems,and to collapse and reduce dimensionality in high-dimensional data,thereby simplifying the object of research problems.In this paper,we divide the research content into four parts:In the first part,we mainly introduce the research background,purpose and significance of the asymptotic normality of the t-distribution and the compressibility with covariates,as well as the main research content and main conclusions of this article.In the second part,we will study the relationship between the Fisher distribution and the t-distribution,using the relationship between the distribution function and the density function,to prove that the Fisher distribution can get the t-distribution,and by using the Kullback-Leibler distance between the two density functions,Finally,the uniform asymptotic normality of the Fisher distribution is proved.In the third part,the Kullback-Leibler distance between the random matrix T-distribution and the matrix normal distribution density function is estimated through an inequality,and a Berry-Esseen bound of the matrix T-distribution is used to prove the random matrix T-distribution uniformly asymptotic normality and gives the speed at which it converges to the normal distribution of the matrix.In the fourth part,by studying the correlation coefficients and conditional correlation coefficients between multivariate t-distributed random variables,we obtained the collapsibility conditions of the multivariate t-distribution with covariates and the normal distribution under the conditional expectation dependence measure collapsibility conditions.