Asymptotic comparison of negative multinomial and multivariate normal experiments

TL;DR

This paper provides a refined approximation of the negative multinomial distribution by a multivariate normal, establishing bounds on their statistical distance and comparing the experiments asymptotically.

Contribution

It introduces a new local approximation method for the negative multinomial, improving understanding of their asymptotic relationship with multivariate normal distributions.

Findings

Derived a refined approximation using Stirling's formula

Established upper bounds on Hellinger and Le Cam distances

Compared negative multinomial and normal experiments asymptotically

Abstract

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.