AI 中文总结
针对现有谱密度估计方法在波动时变趋势等场景性能下降的问题，本文提出适配紧差分的中心对称核及对应最优差分序列，提升了估计性能并适用于多种统计推断任务。
AI 中文摘要
均值稳健的谱密度与长期方差估计对许多统计推断程序至关重要。然而，现有方法在序列依赖数据呈现波动、时变趋势或突发跳跃时，尤其是小样本情况下，性能常下降。差分与核平均是实现均值稳健性与一致性的标准工具，但二者天生不兼容，结合使用会损害最优性。具体而言，紧差分（一种取小滞后差分以增强局部去趋势的操作）会引入强相关性，扭曲核平均估计量的高阶性质。为解决该不兼容问题，本文提出一类专为与紧差分结合而设计的新型中心对称核。研究表明，序列依赖数据的最优紧差分序列不同于针对独立数据设计的经典序列，且这些提出的最优序列与数据无关，可直接应用无需预拟合。最后，所提估计量在多种统计推断任务中展现出实用性，包括平稳性与白噪声检验。
英文摘要
Mean-robust estimation of spectral density and long-run variance is crucial for many statistical inference procedures. However, existing methods often degrade when serially dependent data exhibit volatile, time-varying trends, or sudden jumps, particularly in small samples. While differencing and kernel averaging are standard tools for achieving mean robustness and consistency, they are not inherently compatible. Combining them can compromise optimality. Specifically, tight differencing, an operation of taking small-lag differences to enhance local de-trending, introduces strong correlations that distort the high-order properties of kernel-averaged estimators. To resolve this incompatibility, we introduce a novel class of centrosymmetric kernels explicitly designed to integrate with tight differencing. We demonstrate that the optimal tight difference sequence for serially dependent data differs from classical sequences designed for independent data. Notably, these proposed optimal sequences are data-independent and can be applied directly without pre-fitting. Finally, the proposed estimators are demonstrated to be useful across various statistical inference tasks, including tests for stationarity and white noise.
