Overview
This research introduces a robust methodology for the identification of continuous-time systems, specifically addressing the challenges posed by narrow-band disturbances in the context of continuous-time ARMA (CARMA) models. Standard maximum-likelihood Whittle estimation, commonly used for such systems, can exhibit sensitivity to affected ordinates, particularly when the high-frequency spectral roll-off of CARMA models amplifies the impact of these disturbances, especially under conditions of weak aliasing. The proposed approach aims to mitigate this sensitivity through a robust spectral estimation technique, grounded in minimax theory.
Research Context
The problem arises in the estimation of CARMA models, where the spectral characteristics, specifically the high-frequency spectral roll-off, can interact with narrow-band disturbances. This interaction leads to an magnified effect of these disturbances on the estimation process when aliasing is weak. Consequently, the performance of standard maximum-likelihood Whittle estimation deteriorates as it becomes sensitive to the specific ordinates impacted by these disturbances. The underlying issue is framed as a scale problem within the Whittle estimation framework.
Approach
The methodology introduces a transformation and a clipping mechanism to achieve robustness. The core steps involve:
- Logarithmic Transformation: A logarithmic transformation, $r_k = \log \rho_k$, is applied to convert the Whittle scale problem into a Gumbel location problem. This establishes a connection between robust spectral estimation and the classical minimax theory, as developed by Huber and Rieder.
- Piecewise Centering Correction: A piecewise centering correction is incorporated. This correction is provided in a closed-form for small clipping levels (denoted as $b$), and as an implicit closed-form for the practitioner-relevant range. The purpose of this correction is to maintain Fisher consistency for any given clipping level without necessitating numerical optimization.
- Two-Sided Clipped Gumbel Score: Symmetrically clipping the Gumbel score and subsequently transforming it back yields a two-sided clipped Gumbel score. This form corresponds to the standard Rieder–Hampel bounded-influence robust estimator.
- Minimax Optimality: The normalized influence curve of this two-sided clipped Gumbel score is demonstrated to be locally asymptotically minimax. This optimality holds under conditions of shrinking gross-error contamination, indicating that the estimator achieves the best possible performance in the worst-case scenario of a small amount of contamination.
Findings
The research yielded several quantitative and qualitative findings:
- Efficiency Loss Quantification: The efficiency loss of the robust method is quantified by a single scalar, $K(b)$, which depends on the clipping level $b$. For a specific clipping level of $b = 1.5$, the nominal asymptotic variance overhead is quantified at $16\%$.
- Compatibility with Asymptotic Rates: In an AR(2) Monte Carlo design, the observed sample-size trends were found to be compatible with the asymptotic rate predicted by the theory.
- Bias Reductions: For the AR(2) Monte Carlo design, at a clipping level of $b = 1.5$, significant bias reductions were observed. Specifically, the reductions were approximately $40\%$ for the parameter $a_1$, $90\%$ for $a_2$, and $96\%$ for the parameter $\lambda$.
Why This Matters
This work addresses a fundamental challenge in the robust identification of continuous-time systems by providing a method that is less susceptible to narrow-band disturbances, which can otherwise significantly degrade the accuracy of standard estimation techniques. By connecting robust spectral estimation to classical minimax theory and offering specific quantifiable improvements in bias reduction, the research provides a more reliable tool for system identification in practical scenarios where such disturbances are present.