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
Sharp local minimax rates for goodness-of-fit testing in multivariate binomial and Poisson families and in multinomials cover
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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 pp and nn i.i.d. samples drawn from an unknown distribution qq, we investigate how large ρ>0\rho>0 should be to distinguish, with high probability, the case p=qp=q from the case d(p,q)ρd(p,q) \geq \rho , where dd denotes a specific distance over probability distributions. We answer this question in the case of a family of different distances: d(p,q)=pqtd(p,q) = \|p-q\|_t for t[1,2]t \in [1,2], where t\|\cdot\|_t is the entrywise t\ell_t norm. Besides being locally minimax-optimal – i.e. characterizing the detection threshold in dependence of the known matrix pp – 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