Bayesian estimation of first-order integer generalized autoregressive models based on the negative binomial thinning operator

Integer-valued time series of the heavy-tailed type are seldom considered, it is often observed that the sequence exhibits the characteristic of heavy tails, which suggests that the tail probabilities cannot be ignored. The generalized Poisson-inverse Gaussian (GPIG) family is extremly adaptable and is often used to model heavy-tailed data. Based on this, we proposed a new count time series model, which we abbreviate as the NBINARGPIG(1) model. This model is based on the negative binomial thinning operator with GPIG innovations. This model is examined for stationarity and ergodicity, with the expressions for marginal mean and variance being supplied. Since there is no explicit expression for the posterior distribution, this paper adopts the Markov Chain Monte Carlo algorithm for Bayesian estimation, and compares with maximum likelihood estimation. To illustrate the robustness of the Bayesian estimation, the model with outliers is used to simulate data contamination. Finally, three concrete examples are presented to further illustrate the effectiveness of the model in handling this type of data and the feasibility of the algorithm in solving such problems.