# Sharp local minimax rates for goodness-of-fit testing in multivariate binomial and Poisson families and in multinomials

### Julien Chhor

CREST-ENSAE, Palaiseau, France### Alexandra Carpentier

Universität Potsdam, Germany

## Abstract

We consider the identity testing problem – or goodness-of-fit testing problem – in multivariate binomial families, multivariate Poisson families and multinomial distributions. Given a known distribution $p$ and $n$ i.i.d. samples drawn from an unknown distribution $q$, we investigate how large $ρ>0$ should be to distinguish, with high probability, the case $p=q$ from the case $d(p,q)≥ρ$ , where $d$ denotes a specific distance over probability distributions. We answer this question in the case of a family of different distances: $d(p,q)=∥p−q∥_{t}$ for $t∈[1,2]$, where $∥⋅∥_{t}$ is the entrywise $ℓ_{t}$ norm. Besides being locally minimax-optimal – i.e. characterizing the detection threshold in dependence of the known matrix $p$ – our tests have simple expressions and are easily implementable.

## Cite this article

Julien Chhor, Alexandra Carpentier, Sharp local minimax rates for goodness-of-fit testing in multivariate binomial and Poisson families and in multinomials. Math. Stat. Learn. 5 (2022), no. 1/2, pp. 1–54

DOI 10.4171/MSL/32