AI 中文总结
本文提出最小范数置信包络（MiNCE）框架，证明其强一致一致性，并扩展至频域平滑谱，实验验证了置信带的收缩性。
AI 中文摘要
最小范数置信包络策略提供了一种非参数方法，用于构造带限函数的非渐近、同时置信区域，利用了再生核希尔伯特空间（RKHS）的理论。尽管这些包络的有限样本覆盖保证已被建立，但其一致性尚未被分析。在本文中，我们研究了这种构造，此处称为最小范数置信包络（MiNCE）框架，并在测量噪声的温和假设下，针对无噪声和有噪声观测模型，建立了所得带的强一致一致性。我们进一步将该公式扩展到频域，推导出平滑谱的非渐近、同时、强一致一致置信带。非参数回归和谱估计中的数值实验实证确认了我们的理论结果，说明了随着样本量的增加，置信包络向目标函数的收缩。
英文摘要
Minimum-norm confidence envelope strategies offer a nonparametric approach to constructing nonasymptotic, simultaneous confidence regions for band-limited functions, exploiting the theory of Reproducing Kernel Hilbert Spaces (RKHS). While the finite-sample coverage guarantees of these envelopes have been established, their consistency has not been analyzed so far. In this paper, we study this construction, here termed the Minimum-Norm Confidence Envelope (MiNCE) framework, and establish the strong uniform consistency of the resulting bands, both for noise-free and noisy observation models, under mild assumptions on the measurement noises. We further extend this formulation to the frequency domain, deriving nonasymptotic, simultaneous, strongly uniformly consistent confidence bands for the smoothed spectra. Numerical experiments in nonparametric regression and spectral estimation empirically confirm our theoretical results, illustrating the contraction of the confidence envelopes toward the target function as the sample size increases.
