On the Le Cam distance between multivariate hypergeometric and multivariate normal experiments

TL;DR

This paper derives bounds on the total variation and Le Cam distance between multivariate hypergeometric and normal distributions, using local approximations and existing bounds, to quantify their statistical experiment similarity.

Contribution

It introduces a local approximation for the log-ratio of hypergeometric and multinomial probabilities and applies it to bound the Le Cam distance between hypergeometric and normal experiments.

Findings

Bounds on total variation between hypergeometric and normal distributions.

An upper bound on the Le Cam distance between the two experiments.

Extension of existing bounds with a new local approximation technique.

Abstract

In this short note, we develop a local approximation for the log-ratio of the multivariate hypergeometric probability mass function over the corresponding multinomial probability mass function. In conjunction with the bounds from Carter (2002) and Ouimet (2021) on the total variation between the law of a multinomial vector jittered by a uniform on $(-1/2,1/2)^d$ and the law of the corresponding multivariate normal distribution, the local expansion for the log-ratio is then used to obtain a total variation bound between the law of a multivariate hypergeometric random vector jittered by a uniform on $(-1/2,1/2)^d$ and the law of the corresponding multivariate normal distribution. As a corollary, we find an upper bound on the Le Cam distance between multivariate hypergeometric and multivariate normal experiments.