Multivariate Mean Vector Test Based On Spatial Median In High-dimensional Elliptical Populations

With the rapid development of modern science and technology,people begin to pay more and more attention to the problem of correlative hypothesis testing under non-multivariate normal distribution,including elliptical symmetric distribution.In multi-variate statistical analysis,the definition and properties of multivariate statistics are the basis of hypothesis testing.The distribution of many important statistics under elliptical distribution is an important research problem in generalized multivariate analysis,and the hypothesis testing under high-dimensional elliptical distribution is also a hot topic in statistical research.Therefore,this paper will adopt the correlation statistical analysis method to study the hypothesis testing of mean vector in a high dimensional elliptical distributed population.In this paper,the hypothesis testing of three population mean vectors with high dimensional elliptical distribution is studied.Based on the theoretical properties of space median under the high-dimensional asymptotic framework,It follows Wang’s(2015)idea of converting high-dimensional data with spatial symbol function and Hotelling~2test statistic method developed by Chen&Qin(2010).It is used to construct the mean vector of Hotelling~2test statistics for three populations based on the spatial median in the case of high dimension.The asymptotic expectation of the test statistics is determined by the new asymptotic expression of the value in the sample space proposed by Li&Xu(2022).By using the martingale difference center limit theorem and the(90)7)(6 method,it is concluded that the test statistic follows the normal distribution asymptotically.Next,the hypothesis testing problem of multi-population mean vector under high dimensional elliptical distribution is extended.Based on the theoretical property of the space median in the high-dimensional elliptical model,a new test statistic is constructed according to the three-population case.By using the martingale difference center limit theorem and the(90)7)(6 method,the new test statistic is proved to be asymptotically normal distribution under the null hypothesis.Finally,the method proposed in this paper is studied numerically,and the validity of the method is demonstrated from the probability and efficiency of the first type of error made by the test statistics under different models.