Negative dependence and stochastic orderings

TL;DR

This paper investigates negative dependence in integer-valued random variables, demonstrating they are stochastically smaller than Poisson variables with the same mean, with applications to entropy, approximation, and concentration.

Contribution

It introduces new stochastic ordering results for negatively dependent variables, generalizing to various distributions and providing novel Poisson approximation techniques.

Findings

Negatively dependent variables are smaller than Poisson with same mean in convex order.

Applications include entropy bounds, Poisson approximation, and concentration inequalities.

Results extend to binomial and mixed Poisson distributions.

Abstract

We explore negative dependence and stochastic orderings, showing that if an integer-valued random variable $W$ satisfies a certain negative dependence assumption, then $W$ is smaller (in the convex sense) than a Poisson variable of equal mean. Such $W$ include those which may be written as a sum of totally negatively dependent indicators. This is generalised to other stochastic orderings. Applications include entropy bounds, Poisson approximation and concentration. The proof uses thinning and size-biasing. We also show how these give a different Poisson approximation result, which is applied to mixed Poisson distributions. Analogous results for the binomial distribution are also presented.