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Generalized Fréchet Bounds for Cell Entries in Multidimensional Contingency Tables

9 August 2017
Caroline Uhler
Donald Richards
ArXiv (abs)PDFHTML
Abstract

We consider the lattice, L\mathcal{L}L, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, n(⋅)n(\cdot)n(⋅), on L\mathcal{L}L. We derive from the supermodularity of n(⋅)n(\cdot)n(⋅) some generalized Fr\'echet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from n(⋅)n(\cdot)n(⋅), and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequality of Ky Fan to derive a new approach to Fr\'echet inequalities for multidimensional contingency tables.

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