Data consisting of samples of probability density functions are increasingly prevalent, necessitating the development of methodologies for their analysis that respect the inherent nonlinearities associated with densities. In many applications, density curves appear as functional response objects in a regression model with vector predictors. For such models, inference is key to understand the importance of density-predictor relationships, and the uncertainty associated with the estimated conditional mean densities, defined as conditional Fréchet means under a suitable metric. Using the Wasserstein geometry of optimal transport, we consider the Fréchet regression of density curve responses and develop tests for global and partial effects, as well as simultaneous confidence bands for estimated conditional mean densities. The asymptotic behavior of these objects is based on underlying functional central limit theorems within Wasserstein space, and we demonstrate that they are asymptotically of the correct size and coverage, with uniformly strong consistency of the proposed tests under sequences of contiguous alternatives. The accuracy of these methods, including nominal size, power and coverage, is assessed through simulations, and their utility is illustrated through a regression analysis of post-intracerebral hemorrhage hematoma densities and their associations with a set of clinical and radiological covariates.

中文翻译：

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密度响应曲线的Fréchet回归的Wasserstein $ F $-检验和置信带

由概率密度函数的样本组成的数据越来越普遍，因此有必要开发用于分析的方法，这些方法应考虑与密度相关的固有非线性。在许多应用中，密度曲线在具有矢量预测变量的回归模型中显示为功能响应对象。对于此类模型，推断对于理解密度与预测变量之间关系的重要性以及与估计的条件平均密度相关的不确定性（确定为合适度量下的条件Fréchet均值）至关重要。使用最佳运输的Wasserstein几何形状，我们考虑了密度曲线响应的Fréchet回归，并开发了整体效应和局部效应的测试，以及估计条件平均密度的同时置信带。这些对象的渐近行为基于Wasserstein空间内的基础功能中心极限定理，并且我们证明了它们在适当大小和覆盖范围内都具有渐近性，并且在连续选择的序列下，这些拟议的检验具有一致的一致性。通过模拟评估了这些方法的准确性，包括标称大小，功率和覆盖范围，并通过对脑内出血后血肿密度及其与一组临床和放射学协变量的关联进行回归分析来说明其效用。在连续选择的序列下，所提出的测试具有一致的一致性。通过模拟评估了这些方法的准确性，包括标称大小，功率和覆盖范围，并通过对脑内出血后血肿密度及其与一组临床和放射学协变量的关联进行回归分析来说明其效用。在连续选择的序列下，所提出的测试具有一致的一致性。通过模拟评估了这些方法的准确性，包括标称大小，功率和覆盖范围，并通过对脑内出血后血肿密度及其与一组临床和放射学协变量的关联进行回归分析来说明其效用。