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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.25205 |
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| _version_ | 1866915962517716992 |
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| author | Zhao, Yao |
| author_facet | Zhao, Yao |
| contents | Functional autoregressive models of order one (FAR(1)) are predominantly estimated by projecting curves onto leading functional principal components and fitting a vector autoregression in score space, requiring a discrete truncation level $K$ chosen by an \emph{ad hoc} variance threshold. We demonstrate via Monte Carlo experiments that the truncation choice is both consequential and highly regime dependent: the optimal $K$ can differ by an order of magnitude across data-generating regimes, while commonly used high variance thresholds (95\%, 99\%) lead to substantial forecast deterioration, inflating error by up to $35 \%$ relative to an oracle benchmark. We propose a Tikhonov-regularized estimator $\widehatΨ_α= \widehat{C}_1(\widehat{C}_0 + αI)^{-1}$ that replaces the discrete truncation choice with a continuous regularization parameter, selected in a data-driven manner.
We establish the convergence rate $n^{-β/(2(β+1))}$ under a source condition with smoothness parameter $β\in (0, 1]$, achieving the saturation rate $n^{-1/4}$ for smoother targets.
Across three contrasting regimes and four sample sizes, the proposed estimator closely tracks the oracle-best FPCA rule and outperforms it in the most challenging wide-spectrum regime, without prior knowledge of the effective operator dimension.
An application to 2{,}735 daily intraday PM10 curves from Vienna confirms a 9.7\% reduction in mean forecast error relative to the popular 80\% threshold and exhibits more stable parameter adaptation across 16 winter seasons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25205 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Functional Autoregression Without Truncation: A Continuous-Regularization Approach Zhao, Yao Methodology 62M10, 62H25 Functional autoregressive models of order one (FAR(1)) are predominantly estimated by projecting curves onto leading functional principal components and fitting a vector autoregression in score space, requiring a discrete truncation level $K$ chosen by an \emph{ad hoc} variance threshold. We demonstrate via Monte Carlo experiments that the truncation choice is both consequential and highly regime dependent: the optimal $K$ can differ by an order of magnitude across data-generating regimes, while commonly used high variance thresholds (95\%, 99\%) lead to substantial forecast deterioration, inflating error by up to $35 \%$ relative to an oracle benchmark. We propose a Tikhonov-regularized estimator $\widehatΨ_α= \widehat{C}_1(\widehat{C}_0 + αI)^{-1}$ that replaces the discrete truncation choice with a continuous regularization parameter, selected in a data-driven manner. We establish the convergence rate $n^{-β/(2(β+1))}$ under a source condition with smoothness parameter $β\in (0, 1]$, achieving the saturation rate $n^{-1/4}$ for smoother targets. Across three contrasting regimes and four sample sizes, the proposed estimator closely tracks the oracle-best FPCA rule and outperforms it in the most challenging wide-spectrum regime, without prior knowledge of the effective operator dimension. An application to 2{,}735 daily intraday PM10 curves from Vienna confirms a 9.7\% reduction in mean forecast error relative to the popular 80\% threshold and exhibits more stable parameter adaptation across 16 winter seasons. |
| title | Functional Autoregression Without Truncation: A Continuous-Regularization Approach |
| topic | Methodology 62M10, 62H25 |
| url | https://arxiv.org/abs/2604.25205 |