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Testing for changes in the error distribution in functional linear models

Author

Listed:
  • Natalie Neumeyer

    (Universität Hamburg)

  • Leonie Selk

    (Universität Hamburg)

Abstract

We consider linear models with scalar responses and covariates from a separable Hilbert space. The aim is to detect change points in the error distribution, based on sequential residual empirical distribution functions. Expansions for those estimated functions are more challenging in models with infinite-dimensional covariates than in regression models with scalar or vector-valued covariates due to a slower rate of convergence of the parameter estimators. Yet the suggested change point test is asymptotically distribution-free and consistent for one-change point alternatives. In the latter case we also show consistency of a change point estimator.

Suggested Citation

  • Natalie Neumeyer & Leonie Selk, 2025. "Testing for changes in the error distribution in functional linear models," Statistical Papers, Springer, vol. 66(2), pages 1-17, February.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:2:d:10.1007_s00362-024-01656-9
    DOI: 10.1007/s00362-024-01656-9
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    References listed on IDEAS

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    1. Natalie Neumeyer & Ingrid Van Keilegom, 2009. "Change‐Point Tests for the Error Distribution in Non‐parametric Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(3), pages 518-541, September.
    2. István Berkes & Robertas Gabrys & Lajos Horváth & Piotr Kokoszka, 2009. "Detecting changes in the mean of functional observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 927-946, November.
    3. Bai, J., 1994. "Stochastic Equicontinuity and Weak Convergence of Unbounded Sequential Empirical Proceses," Working papers 94-07, Massachusetts Institute of Technology (MIT), Department of Economics.
    4. Lajos Horváth & Piotr Kokoszka & Shanglin Lu, 2024. "Variable Selection Based Testing for Parameter Changes in Regression with Autoregressive Dependence," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 42(4), pages 1331-1343, October.
    5. Boente, Graciela & Parada, Daniela, 2023. "Robust estimation for functional quadratic regression models," Computational Statistics & Data Analysis, Elsevier, vol. 187(C).
    6. Lee, Eun Ryung & Park, Byeong U., 2012. "Sparse estimation in functional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 1-17.
    7. Aston, John A.D. & Kirch, Claudia, 2012. "Detecting and estimating changes in dependent functional data," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 204-220.
    8. Michael G. Akritas & Ingrid Van Keilegom, 2001. "Non‐parametric Estimation of the Residual Distribution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(3), pages 549-567, September.
    9. Leonie Selk & Natalie Neumeyer, 2013. "Testing for a Change of the Innovation Distribution in Nonparametric Autoregression: The Sequential Empirical Process Approach," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 770-788, December.
    10. Aue, Alexander & Gabrys, Robertas & Horváth, Lajos & Kokoszka, Piotr, 2009. "Estimation of a change-point in the mean function of functional data," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2254-2269, November.
    11. Alexander Aue & Gregory Rice & Ozan Sönmez, 2018. "Detecting and dating structural breaks in functional data without dimension reduction," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(3), pages 509-529, June.
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