Minimax-Optimal Robust Identification of Continuous-Time Systems: Handling Narrow-Band Disturbances in the Frequency Domain
Abstract
The high-frequency spectral roll-off of continuous-time ARMA (CARMA) models can magnify the effect of narrow-band disturbances when aliasing is weak, making standard maximum-likelihood Whittle estimation sensitive to affected ordinates. We show that a logarithmic transformation rk = log ρk converts the Whittle scale problem into a Gumbel location problem, connecting robust spectral estimation to the classical minimax theory of Huber and Rieder. A piecewise centering correction---closed-form for small b, implicit closed-form for the practitioner range---preserves Fisher consistency for any clipping level without numerical optimisation. Clipping the Gumbel score symmetrically and transforming back yields a two-sided clipped Gumbel score (the standard Rieder--Hampel bounded-influence form) whose normalised influence curve is proved locally asymptotically minimax under shrinking gross-error contamination. The efficiency loss is quantified by a single scalar K(b): at b = 1.5, only 16% nominal asymptotic variance overhead. In the stated AR(2) Monte Carlo design, the sample-size trends are compatible with the asymptotic rate, and at b=1.5 the bias reductions are about 40%, 90%, and 96% for a1, a2, and λ, respectively.
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BibTeX
@misc{gillberg2026minimaxoptimal,
title = {{Minimax-Optimal Robust Identification of Continuous-Time Systems: Handling Narrow-Band Disturbances in the Frequency Domain}},
author = {Jonas Gillberg and Fredrik Gustafsson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23835},
eprint = {2609.23835},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23835},
}
RIS
TY - UNPB
TI - Minimax-Optimal Robust Identification of Continuous-Time Systems: Handling Narrow-Band Disturbances in the Frequency Domain
AU - Jonas Gillberg
AU - Fredrik Gustafsson
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23835
UR - https://arxiv.org/abs/2609.23835
AB - The high-frequency spectral roll-off of continuous-time ARMA (CARMA) models can magnify the effect of narrow-band disturbances when aliasing is weak, making standard maximum-likelihood Whittle estimation sensitive to affected ordinates. We show that a logarithmic transformation r_k = log ρ_k converts the Whittle scale problem into a Gumbel location problem, connecting robust spectral estimation to the classical minimax theory of Huber and Rieder. A piecewise centering correction---closed-form for small b, implicit closed-form for the practitioner range---preserves Fisher consistency for any clipping level without numerical optimisation. Clipping the Gumbel score symmetrically and transforming back yields a two-sided clipped Gumbel score (the standard Rieder--Hampel bounded-influence form) whose normalised influence curve is proved locally asymptotically minimax under shrinking gross-error contamination. The efficiency loss is quantified by a single scalar K(b): at b = 1.5, only 16% nominal asymptotic variance overhead. In the stated AR(2) Monte Carlo design, the sample-size trends are compatible with the asymptotic rate, and at b=1.5 the bias reductions are about 40%, 90%, and 96% for a_1, a_2, and λ, respectively.
ER -