论文标题
PDE表征几何分布函数和分位数
PDE characterisation of geometric distribution functions and quantiles
论文作者
论文摘要
我们表明，在任何欧几里得空间中，都可以通过其几何（或空间）分布函数明确地重建任意概率度量。重建采用（潜在的）线性PDE的形式，其中差分运算符以封闭形式给出。该结果意味着，与对统计深度社区的共同信念相反，几何CDF原则上提供了对所有深度区域的概率内容的精确控制。我们对几何CDF的规律性进行了全面的研究，并表明一般的连续密度不会产生具有足够规律性的几何CDF，以重建密度。令人惊讶的是，我们证明重建在奇数甚至偶数上显示出不同的行为：它在奇数维度中是局部的，并且在偶数方面是完全非局部性的。我们调查了这个问题，并为偶数维度提供了部分对应物，并在奇数维度中建立了球形对称概率定律的几何CDF的一般表示公式。我们提供了从维度2和3中从其几何CDF中重建密度的明确示例。
We show that in any Euclidean space, an arbitrary probability measure can be reconstructed explicitly by its geometric (or spatial) distribution function. The reconstruction takes the form of a (potentially fractional) linear PDE, where the differential operator is given in closed form. This result implies that, contrary to a common belief in the statistical depth community, geometric cdf's in principle provide exact control over the probability content of all depth regions. We present a comprehensive study of the regularity of the geometric cdf, and show that a continuous density in general does not give rise to a geometric cdf with enough regularity to reconstruct the density pointwise. Surprisingly, we prove that the reconstruction displays different behaviours in odd and even dimension: it is local in odd dimension and completely nonlocal in even dimension. We investigate this issue and provide a partial counterpart for even dimensions, and establish a general representation formula of the geometric cdf of spherically symmetric probability laws in odd dimensions. We provide explicit examples of the reconstruction of a density from its geometric cdf in dimension 2 and 3.
