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

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  • Michael Falk

    (University of Würzburg)

  • Gilles Stupfler

    (University of Nottingham)

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.

Suggested Citation

  • Michael Falk & Gilles Stupfler, 2021. "The Min-characteristic Function: Characterizing Distributions by Their Min-linear Projections," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 254-282, February.
    • Handle: RePEc:spr:sankha:v:83:y:2021:i:1:d:10.1007_s13171-019-00184-1
    DOI: 10.1007/s13171-019-00184-1
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    References listed on IDEAS

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    1. Rafael Schmidt & Ulrich Stadtmüller, 2006. "Non‐parametric Estimation of Tail Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 307-335, June.
    2. Einmahl, J.H.J. & Krajina, A. & Segers, J.J.J., 2007. "A Method of Moments Estimator of Tail Dependence," Other publications TiSEM 6ee60ab8-3c01-4bd9-aa5e-7, Tilburg University, School of Economics and Management.
    3. Drees, Holger & Huang, Xin, 1998. "Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function," Journal of Multivariate Analysis, Elsevier, vol. 64(1), pages 25-47, January.
    4. Falk, Michael & Stupfler, Gilles, 2017. "An offspring of multivariate extreme value theory: The max-characteristic function," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 85-95.
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