The Positive Definitiveness Of Covariance Matrix Of Multivariate Discrete Sample

Quantitative analysis has extensively been used in social, economic and managing fields as the development of computing power is increasing. Its characteristic is using discrete data to describe the performance and basic features of organizations under the study. Consequently, it is very important to consider the optimization of sampling plan specifically on the sample size when multivariate discrete data are considered. Studying the positive definitiveness of the covariance matrix from discrete data is important to design the sampling plan and provides the base for the data analysis.However, there have been few outcomes about the positive definitiveness of covariance matrix, most of which have been restricted to the Covariance-matrix of continuous sample. For example, Dykstra proved that the Covariance-matrix of sample which are independent and follow the same normal distribution is positive definite if and only if n > p (where n is the number of samples, p is the number of variables) by multiplying data matrix of sample with an orthogonal matrix on the left side in 1970 [16]. In 1990, Xie and Chen [9] extend the results for the case of independent variables with same distributions. The covariance matrix of continuous sample is positive definite if and only if n > p also by establishing a zero measure group and equivalent transformation of matrix [9]. However, quite few results can been seen for the case of multivariate discrete data.This thesis discussed the positive definitiveness of the covariance matrix of sample whose variables are discrete data by using" p-vector match, I-Linear combination and their properties. The sufficient and necessary conditions of positive definitiveness of a covariance matrix from multivariate discrete data are obtained. In a special case of 2 dimensional variables, the probability of the positive definitiveness of the discrete sample covariance matrix is given in term of sample size and variable dimension. Based on the results, the optimal sample size is provided in this thesis. Finally, the difference of the positive definite conditions for discrete and continuous sample covariance matrix is discussed.This thesis has mainly contributed two results in theory: one is that the covariance matrix is not positive definitive if the number of sampling is least than the number of variables, the other is that the covariance matrix of discrete sample is positive definitive if and only if the sum of every random column vector of data matrix is non I-linear combination. This result has extended the positive definitiveness of covariance matrix of continuous sample in theory. Further more, this thesis has provided a new method for optimizing sampling in theory, and established the probability model for the positive definitiveness of discrete sample by which the number of sampling was optimized.