Two Sample High-Dimensional Covariance Test

• Activity Number: 123 - New Challenges and Opportunities in Nonparametric Statistics
• Type: Topic Contributed
• Date/Time: Monday, July 29, 2019 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Danning Li* and Lingzhou Xue and Xiufan Yu
• Companies: Penn State University and Penn State University and National Institute of Statistical Sciences and Penn State University
• Keywords: High-dimensional; Covariance matrix; Power enhancement

Abstract

We propose new tests for high-dimensional covariance matrices, which are of significant interest in many areas of large-scale inference. Using extreme-value form statistics to test against sparse alternatives and using quadratic form statistics to test against dense alternatives are two popular testing procedures. However, quadratic form statistics suffer from low power against sparse alternatives, and extreme-value form statistics suffer from low power against dense alternatives with small disturbances. It is very im- portant to derive powerful testing procedures against general alternatives (either dense or sparse). Surprisingly, we prove that extreme-value form and quadratic form statistics are asymptotically independent for testing high-dimensional covariance matrices. Using intermediate limiting distributions, we derive explicit rates of uniform convergence for their joint limiting law. Given asymptotic independencies, we introduce the novel Fisher’s combined probability test for high-dimensional covariance matrices. Under the high-dimensional setting, we derive the correct asymptotic size, and prove the consistent power against general alternatives.