AI 中文总结
针对连续时间系统辨识中窄带扰动导致的Whittle估计敏感问题，提出基于Gumbel位置变换和对称截断得分的鲁棒估计方法，实现局部渐近最小最大性，在b=1.5时方差仅增16%，偏差大幅降低。
AI 中文摘要
连续时间ARMA（CARMA）模型的高频谱滚降在混叠较弱时会放大窄带扰动的影响，使得标准的最大似然Whittle估计对受影响的坐标敏感。我们证明对数变换$r_k = \log \rho_k$将Whittle尺度问题转化为Gumbel位置问题，从而将鲁棒谱估计与Huber和Rieder的经典最小最大理论联系起来。分段居中校正——对于小$b$有闭式解，对于实际范围有隐式闭式解——在无需数值优化的前提下，对任何截断水平保持Fisher一致性。对称截断Gumbel得分并变换回来，得到双侧截断Gumbel得分（标准的Rieder–Hampel有界影响形式），其归一化影响曲线在收缩的粗差污染下被证明是局部渐近最小最大的。效率损失由单个标量$K(b)$量化：在$b = 1.5$时，名义渐近方差仅增加$16\\%$。在所陈述的AR(2)蒙特卡洛设计中，样本量趋势与渐近速率一致，且在$b=1.5$时，$a_1$、$a_2$和$\lambda$的偏差减少分别约为$40\\%$、$90\\%$和$96\\%$。
英文摘要
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.
Comments17 pages, 13 figures
