Minimax-Optimal Robust Identification of Continuous-Time Systems: Handling Narrow-Band Disturbances in the Frequency Domain

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 $r_k=\log {\rho }_k$ 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 $16\mathrm{\%}$ 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 $40\mathrm{\%}$, $90\mathrm{\%}$, and $96\mathrm{\%}$ for $a_1$, $a_2$, and $\lambda$, respectively.