On Some Stationary INAR(1) Processes with Compound Poisson Distributions

Abstract

Aly and Bouzar ([2]) used the backward approach in presence of the binomial thinning operator to construct underdispersed stationary first-order autoregressive integer valued (INAR (1)) processes. The present paper is to be seen as a continuation of their work. The focus of this paper is on the development of stationary INAR (1) processes with discrete compound Poisson innovations. We expand on some recent results obtained by several authors for these processes. A number of theoretical results are established and then used to develop stationary INAR (1) processes with compound Poisson innovations with finite mean. We apply our results to obtain in detail important distributional properties of the new models when the innovation follows the Polya-Aeppli distribution, the non-central Polya-Aeppli distribution, the negative binomial distribution, the noncentral negative binomial distribution, the Poisson-Lindley distribution, the Euler-type distribution and the Euler distribution.

Keywords

• integer-valued time series
• binomial thinning operator
• infinite divisibility
• Euler distribution