非平稳时间序列中的稳定区间检验
Testing for Stable Intervals in Non-Stationary Time Series
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中文总结 AI 辅助
针对非平稳时间序列中可能存在科学意义稳定区间的问题，构建带相依局部平稳误差的回归模型检验框架，通过局部线性回归估计与极值近似实现渐近水平控制的相合检验，并经模拟与实际数据验证。
中文摘要 AI 辅助
许多时间序列在整个观测周期内并不平稳，但可能包含具有科学意义的时段，在这些时段内信号在规定容差范围内保持稳定。我们将此问题构建为带相依局部平稳误差的非平稳回归模型中稳定区间的存在性检验。对于由均值函数导出的信号$d$（包括与参考水平的偏差及导数），持续时长$δ$内的稳定性由$d_\infty=\inf_{t\in[0,1-δ]}\sup_{s\in[t,t+δ]}|d(s)|$来刻画。原假设$d_\infty\geΔ$表示不存在长度为$δ$且保持在容差$Δ$内的区间，拒绝该假设则为存在相关稳定时段提供证据。我们通过局部线性回归估计$d$，并构建$d_\infty$的插入式检验。其渐近分布仅由达到极小极大泛函的近极值窗口决定。我们通过极值集将这种局部化形式化，并推导了核估计量在可能收缩的、具有时变长程方差的指标集上的高斯近似与极值近似。所得检验具有相合性且能实现渐近水平控制。该理论还将基于上确界的相关变化检验扩展到时变长程方差和基于导数的假设情形。模拟实验以及在生理和工程时间序列上的应用验证了该方法的有效性。
英文摘要
Many time series are not stable over their full observation horizon, but may contain scientifically meaningful periods during which a signal remains stable up to a prescribed tolerance. We formulate this as an existence test for stable intervals in a non-stationary regression model with dependent, locally stationary errors. For a signal $d$ derived from the mean function, including deviations from reference levels and derivatives, stability over duration $δ$ is encoded by $d_\infty=\inf_{t\in[0,1-δ]}\sup_{s\in[t,t+δ]}|d(s)|$. The hypothesis $d_\infty\geΔ$ states that no interval of length $δ$ remains within the tolerance $Δ$, while rejection provides evidence for the existence of a relevant stable period. We estimate $d$ by local linear regression and construct plug-in tests for $d_\infty$. The asymptotic distribution is determined only by near-extremal windows at which the minimax functional is attained. We formalize this localization through extremal sets and derive Gaussian and extreme value approximations for kernel estimators over possibly shrinking index sets with time-varying long-run variance. The resulting tests are consistent and have asymptotic level control. The theory also extends supremum-based relevant-change tests to time-varying long-run variance and derivative-based hypotheses. Simulations and applications to physiological and engineering time series illustrate the method.
