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

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) \geq \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) = \|p-q\|_t for t \in [1,2] , where \|\cdot\|_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.