Testing for latent structure via the Wilcoxon–Wigner random matrix of normalized rank statistics

Summary This paper considers the problem of testing for latent structure in large symmetric data matrices. The goal is to develop statistically principled methodology that is flexible in its applicability, computationally efficient and insensitive to extreme data variation, thereby overcoming limitations of existing approaches. To this end, we introduce and systematically study certain 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 centring and scaling terms. These asymptotic results enable rigorous parameter-free and distribution-free spectral methodology for addressing two hypothesis-testing problems, namely community detection and principal submatrix detection. Numerical examples illustrate the performance of the proposed approach. Throughout, our findings are juxtaposed with existing results based on the spectral properties of symmetric random matrices with independent entries in signal-plus-noise data settings.