Generalized Frechet Bounds for Cell Entries in Multidimensional Contingency Tables

Abstract

We consider the lattice, L, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, n(.), on L. We derive from the supermodularity of n(.) some generalized Frechet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from 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 Frechet inequalities for multidimensional contingency tables.