Generalized Fr\'echet Bounds for Cell Entries in Multidimensional Contingency Tables

We consider the lattice, $\mathcal{L}$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $n(\cdot )$, on $\mathcal{L}$. We derive from the supermodularity of $n(\cdot )$ some generalized Fr'echet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from $n(\cdot )$, 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.