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Largest magnitude for off-diagonal auto-correlation coefficients in high dimensional framework

Author

Listed:
  • Maxime Boucher

    (Université Libre de Bruxelles
    Université d’Orléans, Université de Tours, CNRS)

  • Didier Chauveau

    (Université d’Orléans, Université de Tours, CNRS)

  • Marguerite Zani

    (Université d’Orléans, Université de Tours, CNRS)

Abstract

This paper studies the coherence of an high-dimensional observations matrix. Specifically, we describe the limiting distribution of the largest magnitude of correlations matrix associated to our data outside a central band which size depends of the sample size. Using the Chen–Stein method, we show the convergence of the normalized coherence towards a Gumbel distribution. We broaden previous results by considering a 3-regime band structure for the off diagonal covariance matrix, where the largest band is composed of asymptotically vanishing coefficients. We provide an hypothesis test on the covariance structure where the alternative shows a clear dichotomy on the vanishing band. Moreover, we provide numerical simulations illustrating the asymptotic behavior of the coherence with Monte-Carlo experiment. We use a splitting strategy computing correlation matrices by blocks in order to avoid the high-dimensional memory issue.

Suggested Citation

  • Maxime Boucher & Didier Chauveau & Marguerite Zani, 2025. "Largest magnitude for off-diagonal auto-correlation coefficients in high dimensional framework," Statistical Papers, Springer, vol. 66(4), pages 1-40, June.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:4:d:10.1007_s00362-025-01693-y
    DOI: 10.1007/s00362-025-01693-y
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    References listed on IDEAS

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    1. Tony Cai, T. & Jiang, Tiefeng, 2012. "Phase transition in limiting distributions of coherence of high-dimensional random matrices," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 24-39.
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    3. Loubaton, Philippe & Rosuel, Alexis & Vallet, Pascal, 2023. "On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime," Journal of Multivariate Analysis, Elsevier, vol. 194(C).
    4. Qiu, Yumou & Chen, Songxi, 2012. "Test for Bandedness of High Dimensional Covariance Matrices with Bandwidth Estimation," MPRA Paper 46242, University Library of Munich, Germany.
    5. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
    6. Ligeralde, Antonio V & Brown, Bryan W, 1995. "Band Covariance Matrix Estimation Using Restricted Residuals: A Monte Carlo Analysis," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 36(3), pages 751-767, August.
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