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Activity Number: 150 - Recent Advances in Nonparametric Statistical Methods for Complex Data
Type: Invited
Date/Time: Monday, July 29, 2019 : 10:30 AM to 12:20 PM
Sponsor: Section on Nonparametric Statistics
Abstract #300288
Title: Detecting Rare and Weak Spikes in Large Covariance Matrices
Author(s): Zheng Tracy Ke*
Companies: Harvard University
Keywords: Cumulative sum; Gaussian proxy model; Higher Criticism; Marchenko-Pastur law; Tracy-Widom law; Phase transition

Given multivariate Gaussian data, we are interested in testing a null hypothesis where the covariance matrix is the identity against an alternative hypothesis where the covariance matrix has a few spiked eigenvalues. We consider a setting where the number of spikes grows as a power of p, and where each spiked eigenvalue is only slightly larger than 1. We discover a phase transition: the two-dimensional phase space that calibrates the spike sparsity and strengths partitions into the Region of Impossibility and the Region of Possibility. In Region of Impossibility, all tests are (asymptotically) powerless in separating the alternative from the null. In Region of Possibility, there are tests that have (asymptotically) full power. We consider a CuSum test, a trace-based test, an eigenvalue-based Higher Criticism test, and a Tracy-Widom test, among which the CuSum test and eigen-HC test are new. We show that the first two tests have asymptotically full power in Region of Possibility. The study requires careful analysis of the L1-distance of our testing problem and delicate Radom Matrix Theory.

Authors who are presenting talks have a * after their name.

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