论文标题
多元二项式和泊松家族中以及多项式的局部局部最小值用于合适性测试
Sharp Local Minimax Rates for Goodness-of-Fit Testing in multivariate Binomial and Poisson families and in multinomials
论文作者
论文摘要
我们考虑多元二项式家庭，多元泊松家庭和多项式分布中的身份测试问题 - 或合适的测试问题。鉴于已知的发行$ p $和$ n $ iid样品从未知分布$ q $中绘制出来，我们调查了$ρ> 0 $应该是多大的，以较高的概率区分case $ p = q $ case $ d（p，q）\ geqρ$，其中$ d $ $ d $表示特定的距离超过了可能性分布。我们在一个不同距离的家庭中回答这个问题：$ d（p，q）= \ | p-q \ | _t $ for $ t \ in [1,2] $中，其中$ \ | \ | \ cdot \ | _t $是entrywise $ \ ell_t $ norm。除了局部最小值 - 最佳 - 即表征已知矩阵$ p $依赖性检测阈值的表征 - 我们的测试具有简单的表达式，易于实现。
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 $\ell_t$ 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.
