Positive Definiteness of the Sample Covariance Matrix of Multi-poisson Distribution

Abstract

Studying the positive definiteness of the covariance matrix of discrete samples helps to determine whether the dimensionality of the samples can be reduced, which is beneficial for optimizing the number of samples and designing optimal plans for sampling surveys. This paper aims to provide a method to determine the variable numbers of the sample subjecting to Poisson distribution. Methods. It is based on the theory of I-linear combination and its properties which are the author’s previous studying results. Results. study shows the covariance matrix of multi-Poisson distribution is positively defined and the probability of the sample covariance matrix of multi-poisson distribution is about 1 when the sample capacity is very large. Conclusion. The dimension size of the sample data matrix of multi-poisson distribution can be reduced when the sample capacity n is no more than the dimension size p.