Testing for latent structure via the Wilcoxon--Wigner random matrix

Music City Center 

This paper considers the problem of testing for latent structure in large symmetric data matrices. The goal here is to develop statistically principled methodology that is flexible in its applicability and insensitive to data variation, thereby overcoming limitations facing existing approaches. To do so, we introduce and systematically study symmetric matrices, called Wilcoxon--Wigner random matrices, whose entries are normalized rank statistics derived from an underlying independent and identically distributed sample of absolutely continuous random variables. These matrices naturally arise as the matricization of one-sample problems in statistics and conceptually lie at the interface of nonparametrics, multivariate analysis, and data reduction. Among our results, we establish that the leading eigenvalue and corresponding eigenvector of Wilcoxon--Wigner random matrices admit asymptotically Gaussian fluctuations with explicit centering and scaling terms. These asymptotic results, which are parameter-free and distribution-free, enable rigorous spectral methodology for addressing two hypothesis testing problems, namely community detection and principal submatrix localization.

Rank statistic

Hypothesis testing

Semicircle distribution

Bai--Yin law

Outlier eigenvalue and eigenvector

Spectral method 

Section on Nonparametric Statistics