Penalized empirical likelihood inference for the GINAR(p) model

Integer-valued time series data are greatly useful in many applications, such as reporting the daily number of patients impacted by an epidemic. When modelling high-order integer-valued time series data, order selection is a difficult task. In this paper, we propose a penalized empirical likelihood (PEL) method for order selection and parameter estimation in generalized pth-order integer-valued autoregressive (GINAR(p)) model. We show that the PEL method in the GINAR(p) model has oracle properties, which means that the PEL estimators identify the model as efficiently as if the true structure of the model was known ahead of time. Furthermore, we present the PEL ratio statistic to test a linear hypothesis of the parameter and demonstrate that it has an asymptotically $\chi 2$ distribution under the null hypothesis. Numerical simulation and real data analysis are carried out to assess the performance our proposed method.