Estimating invertible processes in Hilbert spaces, with applications to functional ARMA processes
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908471251697664 |
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| author | Kühnert, Sebastian Rice, Gregory Aue, Alexander |
| author_facet | Kühnert, Sebastian Rice, Gregory Aue, Alexander |
| contents | Invertible processes are central to functional time series analysis, making the estimation of their defining operators a key problem. While asymptotic error bounds have been established for specific ARMA models on $L^2[0,1]$, a general theoretical framework has not yet been considered. This paper fills in this gap by deriving consistent estimators for the operators characterizing the invertible representation of a functional time series with white noise innovations in a general separable Hilbert space. Under mild conditions covering a broad class of functional time series, we establish explicit asymptotic error bounds, with rates determined by operator smoothness and eigenvalue decay. These results further provide consistency-rate estimates for operators in Hilbert space-valued causal linear processes, including functional MA, AR, and ARMA models of arbitrary order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Estimating invertible processes in Hilbert spaces, with applications to functional ARMA processes Kühnert, Sebastian Rice, Gregory Aue, Alexander Statistics Theory 60G10, 62F12, 62R10 Invertible processes are central to functional time series analysis, making the estimation of their defining operators a key problem. While asymptotic error bounds have been established for specific ARMA models on $L^2[0,1]$, a general theoretical framework has not yet been considered. This paper fills in this gap by deriving consistent estimators for the operators characterizing the invertible representation of a functional time series with white noise innovations in a general separable Hilbert space. Under mild conditions covering a broad class of functional time series, we establish explicit asymptotic error bounds, with rates determined by operator smoothness and eigenvalue decay. These results further provide consistency-rate estimates for operators in Hilbert space-valued causal linear processes, including functional MA, AR, and ARMA models of arbitrary order. |
| title | Estimating invertible processes in Hilbert spaces, with applications to functional ARMA processes |
| topic | Statistics Theory 60G10, 62F12, 62R10 |
| url | https://arxiv.org/abs/2407.12221 |