The Min-characteristic Function: Characterizing Distributions by Their Min-linear Projections

Michael Falk () and Gilles Stupfler
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Michael Falk: University of Würzburg

Sankhya A: The Indian Journal of Statistics, 2021, vol. 83, issue 1, No 11, 254-282

Abstract: Abstract Motivated by a (seemingly previously unnoticed) result stating that d −dimensional distributions on ( 0 , ∞ ) d $(0,\infty )^{d}$ are characterized by the collection of their min-linear projections, we introduce and study a notion of min-characteristic function (min-CF) of a random vector with strictly positive components. Unlike the related notion of max-characteristic function which has been studied recently, the existence of the min-CF does not hinge on any integrability conditions. It is itself a multivariate distribution function, which is continuous and concave, no matter which properties the initial distribution function has. We show the equivalence between convergence in distribution and pointwise convergence of min-CFs, and we also study the functional convergence of the min-CF of the empirical distribution function of a sample of independent and identically distributed random vectors. We provide some further insight into the structure of the set of min-CFs, and we conclude by showing how transforming the components of an arbitrary random vector by a suitable one-to-one transformation such as the exponential function allows the construction of a notion of min-CF for arbitrary random vectors.

Keywords: Characteristic function; Copula; D-norm; Max-linear projections; Min-linear projections; Multivariate distribution.; Primary 60E10; Secondary 62H05 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s13171-019-00184-1

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