This paper is concerned with high-dimensional two-sample mean problems, which receive considerable attention in recent literature. To utilize the correlation information among variables for enhancing the power of two-sample mean tests, we consider the setting in which the precision matrix of high-dimensional data possesses a linear structure. Thus, we first propose a new precision matrix estimation procedure with considering its linear structure, and further develop regularization methods to select the true basis matrices. With the aid of estimated precision matrix, we propose a new statistic for the two-sample mean problems by replacing the inverse of sample covariance matrix in Hotelling test by the estimated precision matrix. The proposed test is applicable for both the low and high dimensional setting. The limiting null distributions of the proposed test statistic under both null and alternative hypotheses are derived. We further derive the asymptotical power function of the proposed test and compare it with existing ones. We found the estimation error of the precision matrix does not have impact on the asymptotical power function. Moreover, asymptotic relative efficiency of the proposed test to the classical Hotelling test tends to infinity when p/n tends to 1. We conduct Monte Carlo simulation study to assess the finite sample performance of precision matrix estimation and the proposed high-dimensional two-sample mean test. Our simulation results imply that our estimation method enjoy low estimation error and the regularization method is able to efficiently select important basis matrices. The proposed test performs well especially when the elements of the vector have unequal variances. We also illustrate the proposed methodology by an empirical analysis of a real-world data set.
About the Speaker:
Runze Li is the Eberly Family Chair Professor in Statistics, The Pennsylvania State University. He served as Co-Editor of Annals of Statistics from 2013 to 2015. Runze Li is Fellow of IMS, ASA and AAAS. His recent honors and awards also include the Distinguished Achievement Award of International Chinese Statistical Association, 2017, Faculty Research Recognition Awards for Outstanding Collaborative Research. College of Medicine, Penn State University in 2018 and Distinguished Mentoring Award, Eberly College of Science, Penn State University in 2023. His research interests include theory and methodology in variable selection, feature screening, robust statistics, nonparametric and semiparameteric regression. His interdisciplinary research aims to promote the better use of statistics in social behavioral research, neural science research and climate studies.
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