PDE characterisation of geometric distribution functions and quantiles

Abstract: 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.