Abstract We start by studying first-order autoregressive negative binomial (NBD) processes. We then compare maximum likelihood and moment based estimators of the parameters of the NBD INAR(1) model and show that the degree of dependence has significant effect on the quality of the estimators. Finally, we construct NBD processes with long-range dependence by using the NBD INAR(1) processes as ba...

Interpretations for the q-binomial coefficient evaluated at −q are discussed. A (q, t)-version is established, including an instance of a cyclic sieving phenomenon involving unitary spaces. 1. The q-binomial The q-binomial coefficient is defined for integers k and n, with 0 ≤ k ≤ n, and an indeterminate q by

Fisher's logarithmic series model (Fisher et al. (1943)) is a classical model in statistical ecology. In this paper we show that this model is a key model linking three models discussed in Takemura (1997), i.e., Poisson-gamma model (Bethlehem et al. (1990)), Dirichlet-multinomial model (Takemura (1997)), and Ewens model (Ewens (1990)). This connection opens up the possibility of applying existi...

Problem statement: The modeling of claims count is one of the most important topics in actuarial theory and practice. Many attempts were implemented in expanding the classes of mixed and compound distributions, especially in the distribution of exponential family, resulting in a better fit on count data. In some cases, it is proven that mixed distributions, in particular mixed Poisson and mixed...