Abstract
We study the sample covariance matrix for real-valued data with general population covariance, as well as MANOVA-type covariance estimators in variance components models under null hypotheses of global sphericity. In the limit as matrix dimensions increase proportionally, the asymptotic spectra of such estimators may have multiple disjoint intervals of support, possibly intersecting the negative half line. We show that the distribution of the extremal eigenvalue at each regular edge of the support has a GOE Tracy–Widom limit. Our proof extends a comparison argument of Ji Oon Lee and Kevin Schnelli, replacing a continuous Green function flow by a discrete Lindeberg swapping scheme.
Funding Statement
ZF was supported in part by a Hertz Foundation Fellowship, an NDSEG Fellowship (DoD AFOSR 32 CFR 168a), and NSF DMS 1916198.
IMJ is supported in part by NIH R01 EB001988, GM134483 and NSF DMS 1407813, 1811614.
Acknowledgments
We are indebted to geneticist Mark Blows for asking the question about Tracy–Widom for random effects models that led to this paper, and for many stimulating discussions. We would like to thank Kevin Schnelli for helpful conversations about [19], and the anonymous referees for helpful comments that improved the paper.
Citation
Zhou Fan. Iain M. Johnstone. "Tracy–Widom at each edge of real covariance and MANOVA estimators." Ann. Appl. Probab. 32 (4) 2967 - 3003, August 2022. https://doi.org/10.1214/21-AAP1754
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