On Cramér–von Mises statistic for the spectral distribution of random matrices

On Cramér–von Mises statistic for the spectral distribution of random matrices

Let FN and F be the empirical and limiting spectral distributions of an N×N Wigner matrix. The Cramér–von Mises (CvM) statistic is a classical goodness-of-fit statistic that characterizes the distance between FN and F in L2-norm. In this paper, we consider a mesoscopic approximation of the CvM stati...

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Bibliographic Details
Published in:The Annals of applied probability Vol. 32; no. 6; p. 4315
Main Authors: Bao, Zhigang, He, Yukun
Format: Journal Article
Language:English
Published: Hayward Institute of Mathematical Statistics 01-12-2022
Subjects:
Approximation
Constraining
Estimating techniques
Goodness of fit
Mathematical analysis
Matrix
Probability distribution
Random variables
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Summary:Let FN and F be the empirical and limiting spectral distributions of an N×N Wigner matrix. The Cramér–von Mises (CvM) statistic is a classical goodness-of-fit statistic that characterizes the distance between FN and F in L2-norm. In this paper, we consider a mesoscopic approximation of the CvM statistic for Wigner matrices, and derive its limiting distribution. In the Appendix, we also give the limiting distribution of the CvM statistic (without approximation) for the toy model CUE.
ISSN:1050-5164
2168-8737
DOI:10.1214/22-AAP1788