Asymptotic Distribution Of The Maximum Interpoint Distance For High-dimensional Data

The asymptotic property of the extreme value statistics is a popular research topic in the fields of probability and statistics.It is used widely in practice.In the case of where the sample dimension p is fixed,the traditional multivariate statistical analysis can solve the problem of the distribution of extreme value statistics well.However,with the coming of Big Data Era,we will encounter many samples of high-dimensional data.The dimensions p of these data are very large,then the traditional method is inapplicable.Therefore,the asymptotic property of the extreme value statistics under high-dimensional data is a meaningful topic in statistics.This thesis mainly considers the asympotic distribution of the maximum interpoint dis-tance for high-dimensional data.Let X1,X2,…,Xn be a random sample coming from a p-dimensional population with independent sub-Gaussian components.Denote the maximum in-terpoint Euclidean distance by Mn=max1≤i<j≤n ‖Xi-Xj‖.When the dimension p=p(n)→∞ as the sample size n→∞,it proves that Mn2 under a suitable normalization asymptotically obeys Gumbel type distribution.The proofs are mainly dependent on the Stein-Chen Poisson approximation method and the moderate deviation of the sum of independent random variables.An application of the theoretical method on dealing with the statistical inference for high-dimensional data is also considered in the thesis,that is,we will apply the proposed method to study the hypothesis testing problem of high-dimensional covariance matrix.By numerical simulation,the empirical size diagram of the proposed method is drawn at a given significance level and compared with the high-dimensional U statistic method.We can see that the theoreti-cal method is efficient on dealing with high-dimensional data.