Multivariate Statistics And High-Order Multivariate Probability Distribution

In recent years,the emergence of high-dimensional data has made the traditional method of matrix data processing no longer applicable,and tensors provide a natural and effective way to describe such data,which brings great convenience for data collection and processing.The work of this paper is mainly as follows:This paper includes three parts: Firstly,we introduce the quadratic random variables,and obtain the expressions of their expectations and variances.We also investigate the properties of matrix quadratic random variables(matrices),including the independence between variables and the characteristics of the distribution of such quadratic variables(matrices)by matrix partitioning.Secondly,based on the above discussion,quadratic variables are further extended to more general homogeneous polynomial ones.The distributions of homogeneous polynomial variable are expressed in terms of tensors so their properties are studied.Thirdly,we extend the classic multivariate linear model to a third-order tensor model,and obtain the parameter estimation and variance estimation by matriculated.Through the discussion of the growth curve model,we generalize the third-order tensor model to the general high-order tensor model,and obtain the estimator of the parameter tensor.