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 (Ann Stat 30(3):708–730, 2002) and Ouimet (J Stat Plan Inference 215:218–233, 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.

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