This paper considers statistical inference procedures for a class of models for positively correlated count variables called $\alpha$-permanental random fields, and which can be viewed as a family of multivariate negative binomial distributions. Their appealing probabilistic properties have earlier been studied in the literature, while this is the first statistical paper on $\alpha$-permanental random fields. The focus is on maximum likelihood estimation, maximum quasi-likelihood estimation and on maximum composite likelihood estimation based on uni- and bivariate distributions. Furthermore, new results for $\alpha$-permanents and for a bivariate $\alpha$-permanental random field are presented.

Keywords: $\alpha$-permanent, $\alpha$-permanental random field, composite likelihood, doubly stochastic construction, maximum likelihood, quasi-likelihood