  This paper investigates the theoretical and empirical performance of
Fisher-Pitman-type permutation tests for assessing the equality of unknown
Poisson mixture distributions. Building on nonparametric maximum likelihood
estimators (NPMLEs) of the mixing distribution, these tests are theoretically
shown to be able to adapt to complicated unspecified structures of count data
and also consistent against their corresponding ANOVA-type alternatives; the
latter is a result in parallel to classic claims made by Robinson (Robinson,
1973). The studied methods are then applied to a single-cell RNA-seq data
obtained from different cell types from brain samples of autism subjects and
healthy controls; empirically, they unveil genes that are differentially
expressed between autism and control subjects yet are missed using common
tests. For justifying their use, rate optimality of NPMLEs is also established
in settings similar to nonparametric Gaussian (Wu and Yang, 2020a) and binomial
mixtures (Tian et al., 2017; Vinayak et al., 2019).
