Bivariate Binomial Moments and Bonferroni-type Inequalities

TL;DR

This paper derives bivariate versions of classical linear inequalities for joint probabilities using binomial moments, improving bounds with increasing parameters and employing combinatorial identities for proofs.

Contribution

It introduces new bivariate bounds for joint probabilities based on binomial moments, enhancing existing inequalities with a combinatorial approach.

Findings

Bounds improve monotonically with increasing parameters

Method reveals a multiplicative structure in the inequalities

Provides explicit bivariate forms of classical inequalities

Abstract

We obtain bivariate forms of Gumbel's, Fr\'echet's and Chung's linear inequalities for $P(S\ge u, T\ge v)$ in terms of the bivariate binomial moments $\{S_{i,j}\}$, $1\le i\le k, 1\le j\le l$ of the joint distribution of $(S,T)$. At $u=v=1$, the Gumbel and Fr\'echet bounds improve monotonically with non-decreasing $(k,l)$. The method of proof uses combinatorial identities, and reveals a multiplicative structure before taking expectation over sample points.