Parsimonious and Efficient Integer-valued Autoregressive Process: Inference and Application

Negative Binomial AutoRegressive-type (NBAR) processes serve as a primary methodological frameworkfor the overdispersed integer-valued time series.However, the factorial terms and the turning parameter inherenting in the probability mass functionrender the model's likelihood function structurally intricate, thereby incurring substantial computational costs.To mitigate this dilemma, a new parsimonious yet efficient integer-valued autoregressive process is proposed,which is specifically built upon the Poisson moment exponential (PME) distribution.As a member of the mixed Poisson family,the PME distribution not only exhibits a more parsimonious structure, but alsoovercomes overdispersionwithout resorting to factorial terms or extra parameters. In this paper, we introduce a class of general PME autoregressive processes, establish their stationarity and ergodicity. For a straightforward illustration of the model's statistical properties, we investigate its linear, approximately linear, and non-linear structural forms.Furthermore, we discuss their conditional maximum likelihood (CML) estimation, establish the asymptotic theory for the CML estimators of the parametersand then analyze their finite-sample behavior by some simulation studies.Last but not least, we validate the outperform of the proposed model by two empirical datasets.