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

TL;DR

This paper explores the mathematical properties of marginal functions in multidimensional contingency tables, deriving generalized inequalities and establishing connections with supermodularity and correlation inequalities.

Contribution

It introduces generalized Fréchet bounds for cell entries, extending existing inequalities, and develops new monotonic and supermodular functions related to contingency tables.

Findings

Derived generalized Fréchet inequalities for contingency tables

Established supermodularity and monotonicity properties of marginal functions

Connected supermodularity with correlation inequalities in probability distributions

Abstract

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.