Some ARMA models for dependent sequences of poisson counts

Abstract

A family of models for discrete-time processes with Poisson marginal distributions is developed and investigated. They have the same correlation structure as the linear ARMA processes. The joint distribution of n consecutive observations in such a process is derived and its properties discussed. In particular, time-reversibility and asymptotic behaviour are considered in detail. A vector autoregressive process is constructed and the behaviour of its components, which are Poisson ARMA processes, is considered. In particular, the two-dimensional case is discussed in detail.

Thinning operations for modeling time series of counts—a survey

“…For a definition and historical development of the latter, refer to Heyde and Seneta (1972). Finally, the INAR(1) model is also related to the M/M/∞ queueing system, see McKenzie (1988). With these interpretations, the INAR(1) model applies well to many situations in practice, see Weiß (2007a).…”

“…As an example, altogether four different INMA(q) models have been proposed in the literature, all having different interpretations and probabilistic properties; see AlOsh and , McKenzie (1988) and Brännäs and Hall (2001). The case of Poisson INMA(q) models was investigated by Weiß (2007b).…”

True integer value time series

“…Given this view of our model, it is quite straightforward to define additional models which have a Skellam marginal distribution, by simply taking the difference between two identical but independent Poisson time series processes. This would include the MA(q), ARMA(1, q), ARMA(p, p − 1) and vector AR(1) models considered by McKenzie (1988). The limitation of these models is that the thinning is still defined in terms of a latent process.…”

“…Many properties of these models are found in McKenzie (1988). In particular, Alzaid and Al-Osh (1987) examine the INAR(1) model, covering basic properties and estimation.…”

Modeling time series of counts with a new class of INAR(1) model

“…Du and Li (1991) studied the existence and ergodic properties as well as the lag p dependence of the INAR model. McKenzie (1988) discussed the INAR model in two dimensions and provided details of higher order autoregressive process with Poisson as the marginal distributions; time-reversibility and asymptotic behaviors were also considered. Joe (1996) presented several stochastic processes with a series of univariate margins in the convolution-closed infinitely divisible distributions, including Gamma, inverse Gaussian, and negative binomial as well as generalized Poisson as the special cases.…”