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

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 $\rho >0$ should be to distinguish, with high probability, the case $p=q$ from the case $d(p,q)\ge \rho$ , where $d$ denotes a specific distance over probability distributions. We answer this question in the case of a family of different distances: $d(p,q)=\Vert p-q{\Vert }_t$ for $t\in [1,2]$, where $\Vert \cdot {\Vert }_t$ is the entrywise ${\ell }_t$ 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.