On the Le Cam Distance Between Multivariate Hypergeometric and Multivariate Normal Experiments

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 [4] and Ouimet [14] 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.

Refined normal approximations for the central and noncentral chi-square distributions and some applications

“…For the interested reader, local approximations akin to Lemma 3.1 were derived for the Poisson, binomial, negative binomial, multinomial, Dirichlet, Wishart and multivariate hypergeometric distributions in (Ouimet, 2021a, Lemma 2.1), (Ouimet, 2022a, Lemma 3.1), (Ouimet, 2021c, Lemma 2.1), (Ouimet, 2021b, Theorem 2.1), (Ouimet, 2022b, Theorem 1), (Ouimet, 2022d, Theorem 1), (Ouimet, 2022c, Theorem 1), respectively. See also earlier references such as Govindarajulu (1965) (based on Fourier analysis results from Esseen (1945)) for the Poisson, binomial and negative binomial distributions, and Cressie (1978) for the binomial distribution.…”