Abstract The problem under study is detecting the presence, and identifying the exact position, of a block C n of “anomalous” observations in a finite sequence of independent random variables X 1 , … , X n . By “anomalous” we mean that the distribution G of the observations in the block C n is possibly different from the “under control” distribution F of the remaining X i ’s. We first propose a nonparametric approach, based on classical goodness of fit (Kolmogorov–Smirnov (KS) and Cramer–von Mises (CvM)) statistics, for the case of real random variables X i . An application in stochastic geometry is outlined. Lastly, we focus on the case where the X i are functional data, that is, trajectories of a stochastic process X = X ( t ) , t ∈ [ 0 , 1 ] . The strategy we follow in this case is to take the functional data to the real line with an appropriate transformation and then using the nonparametric detection/identification methodologies mentioned above. The real valued transformation we propose is the Radon–Nikodym derivative of the “under control” distribution (that assumed for most observations) with respect to another suitably chosen reference distribution.

中文翻译：

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单变量和功能数据的非参数检测

摘要 所研究的问题是在独立随机变量 X 1 , … , X n 的有限序列中检测“异常”观测的块 C n 的存在并确定其确切位置。“异常”是指块 C n 中观测值的分布 G 可能不同于其余 X i 的“受控”分布 F。我们首先提出了一种基于经典拟合优度（Kolmogorov-Smirnov (KS) 和 Cramer-von Mises (CvM)）统计的非参数方法，用于真实随机变量 X i 的情况。概述了随机几何中的应用。最后，我们关注 X i 是函数数据的情况，即随机过程的轨迹 X = X ( t ) , t ∈ [ 0 , 1 ] 。在这种情况下，我们遵循的策略是通过适当的转换将功能数据带到实线，然后使用上述非参数检测/识别方法。The real valued transformation we propose is the Radon–Nikodym derivative of the “under control” distribution (that assumed for most observations) with respect to another suitably chosen reference distribution.