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FréChet-Bounds and Their Applications

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Advances in Probability Distributions with Given Marginals

Part of the book series: Mathematics and Its Applications ((MAIA,volume 67))

Summary

This paper gives a review of Fréchet-bounds and their applications. In section two an approach to the marginal problem and Fréchet-bounds based on duality theory resp. the Hahn-Banach theorem is discussed. Main applications concern the Strassen representation theorem for stochastic orders, the sharpness of the classical Fréchet-bounds, the representation of minimal metrics, couplings of distributions, the Monge-Kantorovic-problem, the construction of random variables with maximum (resp. minimum) sum and variances of the sum, maximally dependent random variables and others. For multivariate marginal systems there is a useful reduction principle and there are some bounds for simple systems, which yield a characterization of the marginal problem for a system of two dimensional marginals in a three-fold product space. In section three we discuss some generalizations of the Younginequality, which are useful for solving the dual problems of the Fréchet-bounds. A basic notion in this connection is the notion of c-convex functions. As an application one can give a nice characterization of solutions of certain transportation problems. We give a probabilistic proof of some generalizations of the Young- and the Oppenheim-inequality. In section four we discuss some statistical applications and problems. The Huzurbazar conjecture on marginal sufficiency, the problem of the optimal combination of marginal tests and the question of estimation theory in marginal models is considered.

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  1. Inst, für Math. Statistik, Einsteinstr. 62, D-4400, Münster, Germany

    Ludger Rüschendorf

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  1. Ludger Rüschendorf
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  1. Department of Statistics, Probability and Statistical Applications, University “La Sapienza”, Rome, Italy

    G. Dall’Aglio  & G. Salinetti  & 

  2. Department of Management Science and Statistics, University of Maryland, College Park, USA

    S. Kotz

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Rüschendorf, L. (1991). FréChet-Bounds and Their Applications. In: Dall’Aglio, G., Kotz, S., Salinetti, G. (eds) Advances in Probability Distributions with Given Marginals. Mathematics and Its Applications, vol 67. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3466-8_9

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