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Statistical inference for intrinsic wavelet estimators of SPD covariance matrices in a log-Euclidean manifold

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  • Krebs, Johannes
  • Rademacher, Daniel
  • von Sachs, Rainer

Abstract

In this paper we treat statistical inference for an intrinsic wavelet estimator of curves of symmetric positive definite (SPD) matrices in a log-Euclidean manifold. This estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups. The core of our work is the proposition of confidence sets for our high-level wavelet estimator in a non-Euclidean geometry. We derive asymptotic normality of this estimator, including explicit expressions of its asymptotic variance. This opens the door for constructing asymptotic confidence regions which we compare with our proposed bootstrap scheme for inference. Detailed numerical simulations confirm the appropriateness of our suggested inference schemes.

Suggested Citation

  • Krebs, Johannes & Rademacher, Daniel & von Sachs, Rainer, 2022. "Statistical inference for intrinsic wavelet estimators of SPD covariance matrices in a log-Euclidean manifold," LIDAM Discussion Papers ISBA 2022004, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2022004
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    References listed on IDEAS

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    1. Chau, Van Vinh & von Sachs, Rainer, 2016. "Functional mixed effects wavelet estimation for spectra of replicated time series," LIDAM Discussion Papers ISBA 2016013, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Härdle, Wolfgang & Huet, Sylvie & Mammen, Enno & Sperlich, Stefan, 2004. "Bootstrap Inference In Semiparametric Generalized Additive Models," Econometric Theory, Cambridge University Press, vol. 20(2), pages 265-300, April.
    3. Chau, Van Vinh & von Sachs, Rainer, 2016. "Functional mixed effects wavelet estimation for spectra of replicated time series," LIDAM Reprints ISBA 2016025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Ying Yuan & Hongtu Zhu & Weili Lin & J. S. Marron, 2012. "Local polynomial regression for symmetric positive definite matrices," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(4), pages 697-719, September.
    5. Zhu, Hongtu & Chen, Yasheng & Ibrahim, Joseph G. & Li, Yimei & Hall, Colin & Lin, Weili, 2009. "Intrinsic Regression Models for Positive-Definite Matrices With Applications to Diffusion Tensor Imaging," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 1203-1212.
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    Cited by:

    1. Bailly, Gabriel & von Sachs, Rainer, 2024. "Time-Varying Covariance Matrices Estimation by Nonlinear Wavelet Thresholding in a Log-Euclidean Riemannian Manifold," LIDAM Discussion Papers ISBA 2024004, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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