Recovering a probability measure from its multivariate spatial rank
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We address the problem of recovering a probability measure P over ^n (e.g. its density f_P if one exists) knowing the associated multivariate spatial rank R_P only. It has been shown in <cit.> that multivariate spatial ranks characterize probability measures. We strengthen this result by explictly recovering f_P by R_P in the form of a (potentially fractional) partial differential equation f_P = _n (R_P), where _n is a differential operator given in closed form that depends on n. When P admits no density, we further show that the equality P=_n (R_P) still holds in the sense of distributions (i.e. generalized functions). We throughly investigate the regularity properties of spatial ranks and use the PDE we established to give qualitative results on depths contours and regions. illustrate the relation between f_P and R_P on a few examples in dimension 2 and 3. We study the local properties of the operator _n and show that it is non-local when n is even. We conclude the paper with a partial counterpart to the non-localizability in even dimensions.
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