## generalised Poisson difference autoregressive processes

**Y**esterday, Giulia Carallo arXived the paper on generalised Poisson difference autoregressive processes that is a component of her Ph.D. thesis at Ca’ Foscari Universita di Venezia and to which I contributed while visiting Venezia last Spring. The stochastic process under study is integer valued as a difference of two generalised Poisson variates, made dependent by an INGARCH process that expresses the mean as a regression over past values of the process and past means. Which can be easily simulated as a difference of (correlated) Poisson variates. These two variates can in their turn be (re)defined through a thinning operator that I find most compelling, namely as a sum of Poisson variates with a number of terms being a (quasi-) Binomial variate depending on the previous value. This representation proves useful in establishing stationarity conditions on the process. Beyond establishing various properties of the process, the paper also examines how to conduct Bayesian inference in this context, with specialised Gibbs samplers in action. And comparing models on real datasets via Geyer‘s (1994) logistic approximation to Bayes factors.

February 14, 2020 at 8:53 pm

These generalized Poisson distributions are also called Abel distributions page 17 in ‘Natural exponential families with cubic variance functions’ Annals of Statistics Vol 18 pages 1-37. A beautiful property is that if $S_n=X_1+cdots+X_n$ is the sum of iid Abel-gP variates, and for any $k$ and $T_k=\inf\{\n; n-S_n=k}$ then

$T_k-k$ is Abel-gP. It seems a challenging problem to find the variance function of the exponntial family generated by the law of $X-Y$ where $X$ and $Y are Abel gP, while it is easy when they are only Poisson.

February 15, 2020 at 9:05 am

Sorry for missing the reference, Gérard!