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 $n$ iid 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 $d (p, q) \geq ρ$ , 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 $t \in [1, 2]$ where $∥ \cdot ∥_{t}$ 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.

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 $n$ iid 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 $d (p, q) \geq ρ$ , 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 $t \in [1, 2]$ where $∥ \cdot ∥_{t}$ 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.

文献摘要