Covariance Tensor And Tensor Representation Of Higher Order Moments

Matrix theory is one of the most basic tools in mathematics and science,which facilitates the collection and processing of data.Due to the widespread application of multiindex datasets in the fields of science and engineering,the concept of tensor or hyperfield which is the higher-order generalization of the matrix has been introduced and studied.There are more subscripts for tensors than matrices that have their own geometric and algebraic structures that may be lost if reshaped or expanded into matrices.More complex than the case of matrices is that one of the intrinsic features of them is the concept of tensor eigenvalues,which is very dependent on the tensor structure.Therefore,the tensor itself must be regarded as a data object,and a theory about this new object is currently needed.This paper mainly discusses the tensor representation of covariance tensor and higherorder moment,and the specific contents are as follows: in chapter 3,firstly,the block characteristics of covariance matrix of conditional normal distribution random variables are introduced;then,the linear regression model corresponding to singular covariance matrix and the block method of matrix are studied through the block method of matrix,matrix decomposition and generalized inverse of matrix;finally,the covariance matrix is extended to covariance tensor and its properties are studied.In chapter 4,we first introduce high-order tensor moment generating functions and cumulative tensors;secondly,for random matrix sequences of the same size,we examine the properties of their corresponding tensor moment generating functions and cumulative tensors;finally,we introduce higher-order tensor moment,give the tensor representation of higher-order moment of multivariate Gaussian distribution,and study their properties.In chapter 5,firstly,BVGW distribution is introduced,and its moment generating function and k-order moment expressions are derived.By normalizing the sample points,its approximate distribution function,approximate probability density function and approximate survival function are obtained;on this basis,the EM algorithm is used to calculate the maximum likelihood estimation of unknown parameters in the two-parameter generalized Rayleigh distribution,and the observed Fisher matrix is obtained;according to the maximum likelihood estimation of the parameters asymptotically obey the normal distribution,thus constructing the asymptotic confidence interval of the unknown parameters;secondly,we study the Fisher matrix of the vector VAR process and the application of the Fisher matrix in the Cramer-Rao inequality and Jeffreys non-information prior;finally based on the Fisher matrix with good properties such as symmetry and non-negativity,the Fisher tensor is introduced,and the covariance tensor of the asymptotic distribution of the parameter maximum likelihood estimator is the inverse of the Fisher tensor.At the same time,the properties of the Fisher tensor are discussed.