| With the progress of sensing technology,storage technology and transmission technology,the way of data collection has completely changed,which has promoted the arrival of the era of big data.With the rapid development of advanced technology,highdimensional data has emerged in more and more fields.In the era of big data,data shows new features such as fast collection speed,large dimensions and samples,and complex structure.These new features present unprecedented challenges to the statistical analysis of data,or lead to the problem that there is not defined in traditional statistics,or the existing data cannot be solved.At this point,in order to more accurately reflect the characteristics of observed data,the design of statistical models is becoming increasingly complex,and the inference methods for data are also becoming more and more extensive.Therefore,traditional statistical hypothesis testing methods may not meet the definition of statistics or the efficacy and nature of classical test statistics are not ideal.Under the background of such high-dimensional data,there are many statistical inference methods of hypothesis testing,such as large-scale multiple testing methods,high-dimensional mean vector testing methods,high-dimensional mean vector testing methods under the spike model,high-dimensional covariance matrix testing methods,high-dimensional covariance matrix testing methods under the spike model,and highdimensional mean vector and covariance matrix simultaneous testing methods.This paper proposes a new hypothesis testing method and a new hypothesis testing statistic for high-dimensional data.At the same time,benefiting from the advantages and flexibility of hypothesis testing methods,certain assumptions of existing models can also be further relaxed.The main contents of this paper are as follows:(1)We study the homogeneity test of k population covariance matrices with spike covariance structure in high-dimensional data.We first provide the definition and characteristics of the spike covariance structure,and then propose a test statistic for the covariance matrix under the spike model.We use the noise reduction and the cross data matrix methods to modify the proposed test statistic to obtain a new test statistic.Under the null hypothesis,the asymptotic distribution of the new test statistic was obtained.Theoretical and simulation results indicate that the proposed testing method performs better than existing testing methods.Finally,a real data analysis also proves the superiority of the proposed test method.(2)A new test statistic is proposed for the homogeneity problem of the k-sample covariance matrix based on the Frobenius norm in high-dimensional data.We first give the null hypothesis and the alternative hypothesis based on the weighted Frobenius norm,then propose the test statistics based on the weighted Frobenius norm and test the homogeneity of the covariance matrix of the k sample.Under the null hypothesis and alternative hypothesis,the asymptotic distributions of the new test statistics are given respectively.The simulation results show that the proposed test method is often superior to some existing test methods.Finally,in the real data analysis,the proposed testing method outperforms some existing testing methods.(3)The problem of simultaneous testing of k population mean vector and covariance matrix for high dimensional data is studied.We first present the hull hypothesis and the alternative hypothesis for simultaneous testing,and then present the test statistics for the mean vector and covariance matrix simultaneous testing problem with highdimensional data.Under the null hypothesis and the alternative hypothesis,the asymptotic distributions of the new test statistics are given.The simulation results show that the proposed test method is often better than some existing test methods.Finally,a real data analysis is given to prove the superiority of the proposed method. |
