On Periodic Generalized Poisson INAR(p) Models

Abstract

This paper deals with some probabilistic and statistical properties of periodic generalized Poisson integer-valued autoregressive processes of order p, $PG\mathcal{P}\mathit{INAR}(p).$ Necessary and sufficient conditions for the periodic stationarity, both in mean and second order, are established. The closed-forms of the mean and the second moment are, under these conditions, obtained. Moreover, the Wold–Cramér expression of the underlying second-order periodically stationary process is then established. The autocovariance structure is studied, while providing the closed-form of the periodic autocorrelation function. The Yule–Walker (YW), the two-stage conditional least squares (CLS) and the conditional maximum likelihood (CML) estimation methods of the underlying parameters are obtained. An intensive simulation study and an application on a real count data consisting of the daily number of daytime road accidents in Schiphol area, in Netherlands are provided.

Keywords:

• Periodic generalized Poisson
• Periodically correlated integer-valued process
• Periodically stationary model
• Periodic INAR model

AMS Subject Classification:

• 62F12
• 62M10

Acknowledgements

The authors would like to present their most sincere thanks and their deep gratitude to Professor N. Balakrishnan, Editor-in-Chief, for his helps and continued encouragements. Also, they present to the anonymous referee their most sincere thanks and profound acknowledgements for his pleasant assistance, important corrections, constructive remarks and suggestions which have allowed us to improve the quality and the readability of the paper.