Adaptive Testing Methods For One-sample High-dimensional Covariance Matrices

In recent years,the rapid development of computer science and technology has made it possible to collect more data at a lower cost.These massive,high-dimensional data appear in various fields,including computational biology,climatology,geology,neurology,health sciences,economics,and finance.For example,in biomedical re-search,in order to obtain a more accurate diagnosis and prognosis,researchers collect and analyze a large amount of magnetic resonance images(MRI)data;in economics,researchers usually conduct research on high-dimensional and high-frequency finan-cial data to determine an investment portfolio with excess returns;in geology,the study of geological data helps us understand the evolution of the earth's environment,the formation of mineral resources and other natural issues.These data often present a common feature,that is,the number of covariates p is much larger than the size of sample n,and we called it as "high-dimensional data." The emergence of high-dimensional data not only creates golden opportunities for the development of science,but also brings major challenges for statisticians.Therefore,there is an urgent need to produce statistical inference methods suitable for high-dimensional dataAs we all know,the covariance matrix plays a very important role in multivari-ate statistical analysis.It is the main research object in principal component analysis,factor analysis,linear discriminant analysis and other issues.Therefore,studying the covariance matrix can deepen our understanding of the data,and help us make statisti-cal inferences with the population of the interested data.In this paper,we focus on two important high-dimensional covariance matrix testing problems:one is the global test of high-dimensional one-sample covariance matrix;the other is the test of the banded structure of high-dimensional covariance matrix.In response to the above inspection problems,we respectively propose new methods with different ideas.These methods are not only suitable for high-dimensional data,but also for non-normal distributions More importantly,most of the existing testing methods are only adopt to dense alter-native hypotheses or sparse alternative hypotheses,while our proposed methods can adopt to a wider range of alternative hypotheses,including sparse,dense,etc.First,we conduct research on testing whether the high-dimensional covariance matrix is equal to a given one.Most statisticians use the Frobenius norm of the dif-ference between the identity matrix and the ratio of the covariance matrix to a given positive definite matrix to measure the difference between the covariance matrices un-der the null and alternative hypotheses.However,as so far,no one used the Frobenius norm of the difference between the covariance matrix and the given positive definite matrix as a measure of the alternative hypothesis relative to the null hypothesis for this test issue.But in fact,for these two different types of Frobenius norm-based tests,one of them will not have an absolute advantage over the other.Therefore,we propose a new statistic by combining these two types of statistics.This statistic is suitable for a wider range of dense alternative hypotheses compared to the above single statistic.But when the alternative hypothesis is sparse,the power of the above statistics will be greatly reduced.Nevertheless,the extreme value statistics can solve this problem well.By combining the advantages of these three types of statistics,we finally propose a statistic which is suitable for the alternative hypotheses,including dense,sparse or mixed.Second,we focus on testing whether the high-dimensional covariance matrix has a banded structure.In high-dimensional statistical analysis,since the sample covari-ance matrix is no longer a consistent estimator of the population covariance matrix,people can estimate the population covariance matrix by banding the sample covari-ance matrix.In order to ensure the rationality and validity of this type of estimation,we focus on testsing whether the high-dimensional covariance matrix is banded.Fo-cusing on this hypothesis testing,we propose a series of U statistics based on the La norm of the interested parameter vectors,and obtain its joint asymptotic normality and asymptotic independence under the null hypothesis.By incorporating the independent p-values with minimum p-value method and Fisher's method,we finally propose two adaptive test methods.The extensive simulation results show that the test methods we proposed maintain high power against various alternative hypotheses.