Limit theorems for the Multiplicative Binomial Distribution (MBD)

TL;DR

This paper investigates the asymptotic behavior of the Multiplicative Binomial Distribution (MBD) for diverging parameters and explores how the joint success probability relates to individual Bernoulli responses based on their association.

Contribution

It provides new theoretical insights into the asymptotic properties of the MBD and characterizes the influence of dependence structure on joint success probabilities.

Findings

Asymptotic behavior of MBD as parameters diverge

Relationship between joint and individual success probabilities

Dependence on sign and strength of association

Abstract

The sum of $n$ {non-independent} Bernoulli random variables could be modeled in several different ways. One of these is the Multiplicative Binomial Distribution (MBD), introduced by Altham (1978) and revised by Lovison (1998). In this work, we focus on the distribution asymptotic behavior as its parameters diverge. In addition, we derive a specific property describing the relationship between the joint probability of success of $n$ binary-dependent responses and the individual Bernoulli one; particularly, we prove that it depends on both the sign and the strength of the association between the random variables.