Stationary Underdispersed INAR(1) Models based on the Backward Approach
Keywords:integer-valued time series, binomial thinning operator, q-series, Poissonian Binomial distribution, Heine distribution
Most of the stationary first-order autoregressive integer-valued (INAR(1)) models in the literature have been developed using the idea of binomial thinning. Two approaches have been adopted to establish the distributional properties of a stationary INAR(1) process: the forward approach and the backward approach. In the forward approach the marginal distribution of the processs is specified and an appropriate distribution for the innovation sequence is sought. Whereas in the backward setting, the roles are reversed. The common distribution of the innovation sequence is specified and the marginal distribution of the process is studied. In this article we focus on the backward approach. Our motivation is mainly theoretical, in the context of statistical distribution theory. We establish a number of basic properties of a specific infinite convolution of distributions on Z+. We then proceed to interpret our results in the context of stationary INAR(1) models whose innovation has a finite mean. As an application, we present new distributional properties for some stationary INAR(1) models that show underdispersion, including two new INAR(1) models with q-series innovation distributions.
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