We define extrapolation as any type of statistical inference on a conditional function (e.g., a conditional expectation or conditional quantile) evaluated outside of the support of the conditioning variable. This type of extrapolation occurs in many data analysis applications and can invalidate the resulting conclusions if not taken into account. While extrapolating is straightforward in parametric models, it becomes challenging in nonparametric models. In this work, we extend the nonparametric statistical model to explicitly allow for extrapolation and introduce a class of extrapolation assumptions that can be combined with existing inference techniques to draw extrapolation-aware conclusions. The proposed class of extrapolation assumptions stipulate that the conditional function attains its minimal and maximal directional derivative, in each direction, within the observed support. We illustrate how the framework applies to several statistical applications including prediction and uncertainty quantification. We furthermore propose a consistent estimation procedure that can be used to adjust existing nonparametric estimates to account for extrapolation by providing lower and upper extrapolation bounds. The procedure is empirically evaluated on both simulated and real-world data.

翻译：我们将“外推”定义为对条件函数（例如条件期望或条件分位数）在其条件变量支撑集之外进行的任何类型的统计推断。这种外推发生在许多数据分析应用中，如果未被考虑，可能会使结论失效。尽管在参数模型中外推是直接的，但在非参数模型中变得具有挑战性。在本工作中，我们扩展了非参数统计模型以显式允许外推，并引入了一类外推假设，这些假设可以与现有的推断技术结合以得出具有外推意识的结论。所提出的外推假设规定，条件函数在每个方向上的最小和最大方向导数均在观测到的支撑集内达到。我们展示了该框架如何应用于包括预测和不确定性量化在内的若干统计应用。此外，我们提出了一种一致的估计程序，可通过提供下界和上界外推界来调整现有的非参数估计以考虑外推。该程序在模拟数据和真实数据上均进行了实证评估。