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On the rate of convergence for the autocorrelation operator in functional autoregression

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  • Caponera, Alessia
  • Panaretos, Victor M.

Abstract

We consider the problem of estimating the autocorrelation operator of an autoregressive Hilbertian process. By means of a Tikhonov approach, we establish a general result that yields the convergence rate of the estimated autocorrelation operator as a function of the rate of convergence of the estimated lag zero and lag one autocovariance operators. The result is general in that it can accommodate any consistent estimators of the lagged autocovariances. Consequently it can be applied to processes under any mode of observation: complete, discrete, sparse, and/or with measurement errors. An appealing feature is that the result does not require delicate spectral decay assumptions on the autocovariances but instead rests on natural source conditions. The result is illustrated by application to important special cases.

Suggested Citation

  • Caponera, Alessia & Panaretos, Victor M., 2022. "On the rate of convergence for the autocorrelation operator in functional autoregression," Statistics & Probability Letters, Elsevier, vol. 189(C).
    • Handle: RePEc:eee:stapro:v:189:y:2022:i:c:s0167715222001274
    DOI: 10.1016/j.spl.2022.109575
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    1. Aneiros, Germán & Horová, Ivana & Hušková, Marie & Vieu, Philippe, 2022. "On functional data analysis and related topics," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
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    4. Yao, Fang & Muller, Hans-Georg & Wang, Jane-Ling, 2005. "Functional Data Analysis for Sparse Longitudinal Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 577-590, June.
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    6. Guillas, Serge, 2001. "Rates of convergence of autocorrelation estimates for autoregressive Hilbertian processes," Statistics & Probability Letters, Elsevier, vol. 55(3), pages 281-291, December.
    7. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.
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