Detecting practically significant dependencies in metric spaces via distance correlations
利用距离相关检测度量空间中的实际显著依赖AI 解读 ↓
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利用距离相关检测度量空间中的实际显著依赖
- 研究问题
- 比起检验完全独立，如何判断两个度量空间随机变量的依赖是否超过具有实际意义的阈值，并在观测存在序列依赖时控制第一类错误？
- 方法与思路
- 作者以距离相关不超过预设阈值为原假设，基于序贯距离相关过程的函数极限定理构造枢轴检验，并推导数据驱动的可拒绝阈值。
- 材料陈述的贡献
- 摘要称，方法适用于特定度量空间的绝对正则过程，可控制阈值检验的第一类错误，并能无需重抽样构造距离相关的置信区间。
- 阅读关注点
- 应核对强负型空间与绝对正则性的具体要求、枢轴化过程和阈值解释，以及实际显著性标准如何设定、置信区间如何构造。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收均未核验；摘要未说明完整依赖和矩条件，阈值需结合应用设定，距离相关也不能直接解释为因果关系。
原文摘要
We take a different look at the problem of testing the independence of two metric-space-valued random variables using the distance correlation. Instead of testing if the distance correlation vanishes exactly, we are interested in the hypothesis that it does not exceed a certain threshold. Our testing problem is motivated by the observation that in many cases it is more reasonable to test for a practically significant dependency since it is rare that a hypothesis of perfect independence is exactly satisfied. This point of view also reflects statistical practice, where one often classifies the strength of the association in categories such as `small', `medium' and `large' and the precise definitions depend on the specific application. To address these problems we develop a pivotal test for the hypothesis that the distance correlation between two random variables does not exceed a pre-specified threshold $Δ$. We also determine a minimum value $\hat Δ_α$ from the data such that the hypothesis is rejected for all $Δ\leq \hat Δ_α$ at controlled type I error $α$. This quantity can be interpreted as a measure of evidence against the hypothesis that the distance correlation is less or equal than $Δ$. The new test is applicable to processes taking values in separable metric spaces of strong negative type, covering Euclidean as well as functional data. We do not assume independent observations, and instead prove our results for absolutely regular sample generating processes, which includes many time series such as ARMA and GARCH models. Our approach is based on a new functional limit theorem for the sequential distance correlation process, and can also be used to construct confidence intervals for the distance correlation without the need for resampling.