Distribution function is essential in statistical inference, and connected with samples to form a directed closed loop by the correspondence theorem in measure theory and the Glivenko-Cantelli and Donsker properties. This connection creates a paradigm for statistical inference. However, existing distribution functions are defined in Euclidean spaces and no longer convenient to use in rapidly evolving data objects of complex nature. It is imperative to develop the concept of distribution function in a more general space to meet emerging needs. Note that the linearity allows us to use hypercubes to define the distribution function in a Euclidean space, but without the linearity in a metric space, we must work with the metric to investigate the probability measure. We introduce a class of metric distribution functions through the metric between random objects and a fixed location in metric spaces. We overcome this challenging step by proving the correspondence theorem and the Glivenko-Cantelli theorem for metric distribution functions in metric spaces that lie the foundation for conducting rational statistical inference for metric space-valued data. Then, we develop homogeneity test and mutual independence test for non-Euclidean random objects, and present comprehensive empirical evidence to support the performance of our proposed methods.
翻译:在统计推断中,分配功能至关重要,并且与样本相关,通过测量理论和格利文科-坎特利和唐斯克特性的对应理论和格利文科-坎特利和唐斯克特性形成一个直接的闭路循环。这种联系为统计推理提供了一个范例。然而,现有的分配功能在欧clidean空间作了定义,不再便于在迅速变化的复杂数据对象中使用。必须在一个更普遍的空间发展分配功能概念,以满足新出现的需要。请注意,线性使我们能够利用超立方体来定义欧西里德空间的分布功能,但是没有测量空间的线性,我们必须与该计量标准合作,以调查概率测量。我们通过随机物体和计量空间固定位置之间的测量标准,引入了一组指标性分布功能。我们克服了这一具有挑战性的步骤,证明对应标的和格利文科-坎特利的参数分配功能,这是对计量空间价值数据进行合理统计推断的基础。然后,我们开发了同质测试和相互独立测试,以证明我们提出的非Edideidean随机物体的实性支持。