The Random Order Of The Normal Distribution Of Matrix Variables

As an important tool in probability and statistics,stochastic orders aroused more and more scholars' attention.Stochastic orders had became one of the most popular researches because of its identity,which people could use to choose the better method.Stochastic orders had been widely applied in survival analysis,economics,operations research,biomathemat-ics,insurance actuarial science and other related fields.As a crucial branch of probability distributions,stochastic orders had been defined as a binary relationship,because we always use the nature of stochastic orders to compare random variables so as to choose one which can count for much.In this thesis,we apply the stochastic orders to the normal distribution of matrix variables in order to find the necessary and sufficient conditions between of matrix variable normal distribution and stochastic orders.Stochastic orders provide more convenient methods into the comparison of two random variables than only through their means and variances,which may not exist.According to the relationship between matrix variables and multivariate variables,we extend stochastic order relations necessary and sufficient on multivariate normal distribution to matrix variable normal distribution.In the process of proof,we use a property of Ef(Y)-Ef(X),whereX,Y are matrix variables of normal distribution,f is a function that satisfies some weak regularity conditions.According to this property,we can get the condition that the variables satisfy when we know stochastic order relations.On the contrary,if the relationship between the random variables is known,we can deduce which stochastic orders hold.The first chapter in this thesis mainly introduces the research background of stochastic orders and the development of present situation.We also introduce the means of various sym-bols that appear in the article.In the second chapter,we mainly introduce the definition of matrix variable normal distribution and some classical stochastic orders as well as the lemma that used in this thesis.We also proved that the characteristic functions of matrix variable normal distribution is equal to multivariate normal distribution,when some conditions have been filled.The second chapter also briefly introduces the relationship between multivariate normal distribution and matrix variable normal distribution.A lot of important matrix functions are mentioned in this thesis,in order to facilitate the operation,we prove that the multivariate function and the matrix function are equivalent when some conditions filled so that we can convert the operation of the matrix function into the multivariate function.The third chapter first discusses the nature of Ef(Y)-Ef(X),we use the extension of Plackett proof and the partial integral to prove it,we also describe the weak regularity conditions of this property.And then,this property of the Ef(Y)-Ef(X)is applied to the relation-ship between various random order and normal distribution,some useful results have been obtained.The fourth chapter is a brief summary of the content of this thesis.