对象值时间序列中的长程依赖性量化

中文摘要 AI 辅助

对象值时间序列，包括分布、协方差矩阵、网络和成分数据，缺乏传统基于自协方差的长记忆定义所需的线性结构。我们开发了一个内在框架，用于在负类型度量空间中定义和估计长记忆。一个等距嵌入产生了一个中心化的希尔伯特值过程，其滞后协方差算子的迹提供了时间依赖性的有符号、可加性度量。至关重要的是，该迹等于边际平均成对距离与期望滞后距离之差，因此该框架及其估计器仅使用原始对象之间的距离。我们通过该迹的不可和衰减来定义记忆参数，并表明在希尔伯特值设定下它与通常的参数一致。估计使用基于距离的滞后度量的Bartlett聚合。从相同的相依样本估计共同的边际距离会在这些聚合中引入共同中心化偏差。我们推导了其有限样本形式，并提出了迭代块差分校正、对数比率和多带宽对数斜率估计器，以及局部自洽性细化。我们建立了相合性，在模拟中记录了显著的偏差减少，并在外汇收益分布和美国发电成分中发现了长记忆证据。

英文摘要

Object-valued time series, including distributions, covariance matrices, networks, and compositions, lack the linear structure required by conventional autocovariance-based definitions of long memory. We develop an intrinsic framework for defining and estimating long memory in metric spaces of negative type. An isometric embedding yields a centered Hilbert-valued process, and the trace of its lag-covariance operator provides a signed, additive measure of temporal dependence. Crucially, this trace equals the difference between the marginal mean pairwise distance and expected lagged distance, so the framework and its estimators use only distances between the original objects. We define the memory parameter through the nonsummable decay of this trace and show that it coincides with the usual parameter in Hilbert-valued settings. Estimation uses Bartlett aggregates of distance-based lag measures. Estimating the common marginal distance from the same dependent sample induces a common-centering bias in these aggregates. We derive its finite-sample form and propose iterated block-difference corrections, log-ratio and multi-bandwidth log-slope estimators, and a localized self-consistency refinement. We establish consistency, document substantial bias reduction in simulations, and find long-memory evidence in foreign-exchange return distributions and U.S. electricity-generation compositions.

发表机构

• University of Sydney（悉尼大学）
• Macquarie University（麦考瑞大学）

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