用于上确界型统计量相关推断的自归一化方法
Selfnormalization for relevant inference with supremum-type statistics
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中文总结 AI 辅助
该研究针对函数时间序列相关变化的上确界型统计推断难题，提出结合平滑近似与偏差校正的自归一化方法，得到渐近精确检验，为相关假设下的统计推断提供了新框架。
中文摘要 AI 辅助
我们针对函数时间序列中以上确界范数度量的相关变化推断问题，提出了一种自归一化方法。主要难点在于上确界范数不满足阿达马可微性，因此标准的基于投影的自归一化方法无法适用，且极限分布可能依赖于极值集的几何结构和长期协方差结构。我们通过用平滑的对数-求和-指数（log-sum-exp）近似替代上确界范数，并从其导数构造投影自归一化量来解决该问题。所得统计量具有渐近枢轴分布，不依赖长期协方差冗余参数，仅与断点位置相关。我们针对孤立非退化极值和正测度极值集推导了显式的平滑偏差展开式，为避免直接估计极值的数量、曲率或测度等几何量，我们结合多个平滑水平抵消主导偏差项，在温和正则条件下得到了针对相关变化的渐近精确检验。更广泛地说，所提出的平滑与偏差校正原则为涉及相关假设的问题中自归一化与上确界型统计量的结合提供了框架。
英文摘要
We develop a selfnormalized approach to inference for relevant changes in functional time series measured by the supremum norm. The main difficulty is that the supremum norm is not Hadamard differentiable, so standard projection-based selfnormalization does not apply and the limiting distribution may depend on the geometry of the extremal set and the long-run covariance structure. We address this problem by replacing the supremum norm with a smooth log-sum-exp approximation and constructing a projected selfnormalizer from its derivative. The resulting statistic has an asymptotically pivotal distribution that is free of long-run covariance nuisance parameters and depends only on the break location. We derive explicit smoothing-bias expansions for both isolated nondegenerate extrema and extremal sets of positive measure. To avoid direct estimation of geometric quantities such as the number, curvature, or measure of the extrema, we combine several smoothing levels to cancel the leading bias terms. This yields an asymptotically exact test for relevant changes under mild regularity conditions. More generally, the proposed smoothing and bias-correction principles provide a framework for combining selfnormalization with supremum-type statistics in problems involving relevant hypotheses.
