模糊数据下Tukey深度函数的连续性
On the continuity of the Tukey depth function for fuzzy data
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中文总结 AI 辅助
本文首次系统研究模糊Tukey深度的正则性，证明其关于空间元素的上半连续性与关于分布的经验相合性，确保深度区域封闭并支持样本近似总体，同时推导最深点估计量相合性及有限网格近似。
中文摘要 AI 辅助
统计深度在模糊数据中的实际应用需要正则性、推断稳定性和计算可行性。本文首次针对模糊深度，特别是模糊Tukey深度，研究了这些方面。具体而言，我们探讨了该深度关于其每个参数的连续性。作为底层空间元素的函数，我们证明了在模糊空间中使用的主要度量下其上半连续性。作为计算深度所依据的分布的函数，我们建立了其经验版本对连续模糊随机变量的几乎必然一致相合性。这些结果确保了深度区域的封闭性，并支持使用样本深度值来近似其总体对应值。我们还推导了经验最深点估计量的相合性，并研究了深度的有限网格近似，该近似得到了理论和模拟结果的支持。
英文摘要
The practical use of statistical depth for fuzzy data requires regularity, inferential stability, and computational feasibility. This paper studies these aspects for the first time for the fuzzy depth, in particular for the fuzzy Tukey depth. In particular, we investigate the continuity of this depth with respect to each of its arguments. As a function of the elements of the underlying space, we prove its upper semicontinuity under the main metrics used in fuzzy spaces. As a function of the distribution with respect to which the depth is computed, we establish the almost sure uniform consistency of its empirical version for continuous fuzzy random variables. These results ensure closed depth regions and support the use of sample depth values to approximate their population counterparts. We also derive consistency of empirical deepest-point estimators and study a finite-grid approximation of the depth, supported by theoretical and simulation results.
发表机构
- Universidad de Cantabria（坎塔布里亚大学）
- Universidad de Oviedo（奥维耶多大学）
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