Minimax-Robust Spectral Characteristics

• Minimax-robust spectral characteristics are frequency-domain methods that minimize worst-case error under model uncertainty.
• They extend classical Wiener and Kalman techniques through convex analysis and saddle-point solutions to ensure robustness.
• Applications include robust filtering, detection, and forecasting, effectively mitigating performance degradation from outliers and spectral variability.

A minimax-robust spectral characteristic is a frequency-domain solution to an optimal estimation, filtering, or detection problem that explicitly minimizes the maximal (worst-case) performance degradation—typically in mean-square error or error exponent—over a prescribed class of models or spectral densities. This concept, central in robust statistical signal processing and control, extends classical Wiener–Kolmogorov or Kalman methods to scenarios where spectral densities, covariance structures, or even distributional assumptions are uncertain, leading to estimators or decision rules that are immune to the most adversarial perturbations allowed by the uncertainty model. The construction and analysis of minimax-robust spectral characteristics unify tools from convex analysis, game theory, Hilbert space projection methods, and variational optimization, resulting in saddle-point solutions, explicit robustification procedures (such as “Huberization” or likelihood ratio clipping), and precise characterization of least favorable spectral densities or distributions.

1. Formulation of the Minimax-Robust Problem

The minimax-robust approach begins by acknowledging that in practice, the spectral properties (e.g., the spectral density $f(\lambda)$ of a stationary process) are seldom known exactly. Instead, $f(\lambda)$ is assumed to belong to an admissible set $D$—often described by moment or energy constraints, bounds, or contamination neighborhoods (e.g., $\varepsilon$-contamination, $f$-divergence, density bands). The design objective is then to find an estimator (filter, detector, or predictor) $h$ (with a given spectral characteristic $h(\cdot)$) that minimizes the worst-case risk:

$\min_{h} \max_{f \in D} \Delta(h; f)$

for estimation (or similar sup-inf formulations for detection). $\Delta(h; f)$ here represents the performance metric (e.g., mean-square error for estimation, exponential error exponent for detection).

In detection problems with spectral uncertainty—such as for Gaussian WSS signals in white noise—the minimax error exponent is controlled by the least favorable power spectral density and achieved by a spectral-domain likelihood ratio test tuned to this density (Zhang et al., 2010, Burnashev, 2021).

In linear estimation, with uncertain spectral densities $(f,g)$ for the process and noise, the goal is to find a robust spectral characteristic $h^0(\lambda)$ and (where applicable) identify the least favorable pair $(f^0,g^0)$ solving:

$$\Delta(h; f^0, g^0) \geq \Delta(h^0; f^0, g^0) \geq \Delta(h^0; f, g) \quad \forall \; (f,g) \in D, \;\; h$$

which is an explicit saddle-point property (Luz et al., 2016, Golichenko et al., 2021, Moklyachuk, 25 Jun 2024).

2. Explicit Construction of Minimax-Robust Spectral Characteristics

The optimal estimator or filter is usually constructed in the frequency domain. For stationary sequences, the classic spectral characteristic is

$$h(e^{i\lambda}) = A(e^{i\lambda}) - \varphi^{-1}(\lambda) r(e^{i\lambda}),$$

with $A(e^{i\lambda})$ representing the symbol of the linear functional, $\varphi(\lambda)$ arising from spectral factorization $f(\lambda) = |\varphi(\lambda)|^2$, and $r(e^{i\lambda})$ constructed from the convolution of parameters of the functional and the process (Moklyachuk, 25 Jun 2024). The mean-square error is

$$\Delta(h; f) = \frac{1}{2\pi} \int_{-\pi}^\pi |A(e^{i\lambda}) - h(e^{i\lambda})|^2 f(\lambda) d\lambda.$$

For more elaborate (e.g., periodically correlated, stationary increments, cointegrated, harmonizable stable, or seasonal) models, $h(\lambda)$ takes the form of a rational function or vector-matrix operation involving the process increments, spectral density matrices, and possibly the spectral densities of both process and noise (Luz et al., 2016, Golichenko et al., 2021, Luz et al., 19 Oct 2025).

In the minimax-robust setting, the estimator is still given by the same formula, but the spectral densities $f(\lambda), g(\lambda)$ are replaced by their least favorable counterparts $f^0(\lambda), g^0(\lambda)$.

The least favorable spectral densities are characterized as the argument maximizers in the admissible set $D$ of the error functional for the estimator built for them:

$$(f^0, g^0) = \arg\max_{(f,g) \in D} \Delta(h(f,g); f,g).$$

Explicit solutions may be possible when $D$ is convex and defined by, e.g., moment or $L_1$ ball, or may require variational (Lagrange multiplier) analysis yielding, for example, spectral densities of moving average or autoregressive form satisfying particular boundary constraints (Moklyachuk, 25 Jun 2024, Luz et al., 2016, Masyutka et al., 2021, Luz et al., 19 Oct 2025).

In Gaussian detection, the least favorable spectral density $\phi^*(\omega)$ is identified by a dominance condition ensuring it minimizes Kullback–Leibler “information” and determines the minimax optimal detector (Zhang et al., 2010).

3. Saddle-Point Solutions and Spectral Game Theory

The minimax formulation is a two-player zero-sum game: nature chooses the spectral density (within $D$) to maximize error, while the estimator aims to minimize it. Existence and explicit construction of the saddle point $(h^0, f^0, g^0)$ rely on convexity and compactness properties of the admissible set and linearity of the mappings involved. The existence of such a saddle point ensures that the minimax estimator is robust against any admissible model in $D$, yielding the equality:

$$\Delta(h; f^0, g^0) \geq \Delta(h^0; f^0, g^0) \geq \Delta(h^0; f, g).$$

In detection, a similar saddle point arises: the test based on the least favorable PSD achieves the minimax error exponent, and no other spectral density in $D$ yields a worse worst-case performance (Zhang et al., 2010, Burnashev, 2021). This approach is rigorously extended to sequential and multi-hypothesis settings, where the least favorable distributions may become data-dependent (Fauß et al., 2021).

4. Examples: Filtering, Extrapolation, and Forecasting

Robust Filtering: In Kalman-type filtering, the paper (Ruckdeschel, 2010) demonstrates that, when model uncertainty is cast as an “SO-neighborhood,” the robust filter correction step is Huberized—large innovations are clipped—to limit outlier effects:

$$X_{t|t} = X_{t|t-1} + H_b \left(M^0_t \Delta Y_t\right),$$

with $H_b(v) = v \cdot \min\{1,\, b/|v|\}$ and the threshold $b$ calibrated to balance nominal efficiency and robustness. This clipping directly regularizes the influence of large spectral components introduced by outliers, yielding robustified spectral characteristics and stable filtering properties.

Interpolation/Extrapolation for Stationary Increments and Cointegrated Sequences: The robust spectral characteristic is explicitly given in terms of the process and noise spectral densities and the problem-specific filters (e.g., difference operators for increments, cointegration parameters for cointegrated sequences) (Luz et al., 2016). The least favorable densities satisfy equality/inequality constraints corresponding to the admissible class $D$, e.g., for $L_1$ balls or canonical moment restrictions.

Detection Under Spectral Uncertainty: In hypothesis testing with uncertain covariance or spectral densities, the robust test is tuned to the least favorable element characterized by integral conditions (e.g., dominance) rather than just a worst-case over endpoints, and the error exponent is explicitly given in terms of the least favorable PSD (Zhang et al., 2010, Burnashev, 2021). Under reasonably general conditions (no need for convexity of the admissible set), the robust likelihood ratio test achieves optimality in the minimax sense.

Seasonality, Long Memory, and Cyclostationarity: Robust spectral characteristics for multiseasonal, cyclostationary, or long-memory models utilize vector-valued spectral density formulations, and robustification proceeds via analogous saddle-point and least-favorable constructions. Applications extend to forecasting and filtering in complex seasonal time series where model misspecification is endemic (Luz et al., 2020, Luz et al., 2021, Luz et al., 2023).

5. Implications for Theory and Practice

The minimax-robust spectral characteristics framework leads to estimators and detectors with provable performance guarantees even in the presence of modeling errors, spectral density misspecification, heavy-tailed innovations, or outlier contamination. The saddle-point construction provides both a quantitative robustness bound (with explicit expressions for maximal degradation) and a recipe for filter or test design that is agnostic to specific model realization within the allowed class.

Practical consequences include:

• Robust filters (e.g., rLS) that directly suppress the influence of high-frequency “spikes” due to outliers, stabilizing the spectrum of the update process (Ruckdeschel, 2010).
• Signal detectors in radar, communications, and finance that maintain prescribed error exponents against worst-case (but admissible) spectral or covariance uncertainties (Zhang et al., 2010, Burnashev, 2021).
• Forecasting and extrapolation algorithms that guarantee maximal mean-square error across all densities in $D$, preventing catastrophic error increase due to undetected model drift or seasonal effect perturbations (Luz et al., 2020, Golichenko et al., 2021, Moklyachuk, 25 Jun 2024).
• Explicit characterizations useful for designing robust algorithms in control, time-series analysis, and environmental modeling.

6. Comparative Analysis and Extensions

The minimax-robust approach systematically outperforms or at least matches the classical (nominal) Wiener, Kolmogorov, or Kalman solutions in terms of worst-case risk, especially under spectral uncertainty. However, it may be more conservative—trading some nominal model efficiency for robustness. Formulas for robust spectral characteristics, least favorable densities, and the resulting MSE or error exponent are generally more involved, requiring convex or variational optimization or solution of Lagrange-multiplier equations.

The approach naturally extends to:

• Multivariate and high-dimensional processes, where admissible sets may involve matrix-valued spectral density bands or trace constraints (Golichenko et al., 2021).
• Non-stationary or increment-stationary signals (ARIMA, seasonal ARIMA), with robustification via minimax interpolation or extrapolation (Luz et al., 2016, Luz et al., 19 Oct 2025).
• Periodically correlated (cyclostationary) sequences through vector-valued spectral methods (Dubovets'ka et al., 19 Oct 2025, Luz et al., 2021).
• Spectral problems in continuous time and in the context of differential operators, where minimax principles in spectral gaps lead to robust eigenvalue estimates (Seelmann, 2020).

This comprehensive minimax-robust spectral estimation and detection paradigm is a key pillar of modern robust signal processing under uncertainty, providing a principled, unified framework for safeguarding inference and estimation against adversarial spectral perturbations, model uncertainty, and outliers.