Stationary Time Series Models with Exponential Dispersion Model Margins

Abstract

We consider a class of stationary infinite-order moving average processes with margins in the class of infinitely divisible exponential dispersion models. The processes are constructed by means of the thinning operation of Joe (1996), generalizing the binomial thinning used by McKenzie (1986, 1988) and Al-Osh and Alzaid (1987) for integer-valued time series. As a special case we obtain a class of autoregressive moving average processes that are different from the ARMA models proposed by Joe (1996). The range of possible marginal distributions for the new models is extensive and includes all infinitely divisible distributions with finite moment generating functions, hereunder many known discrete, continuous and mixed distributions.

Poisson-Driven Stationary Markov Models

“…Unlike other existing approaches in the literature, the proposed construction has a useful representation of the underlying transition probability. Although some other constructions, like those in Joe (1996), Jørgensen andSong (1996), andPitt, Chatfield, andWalker (2002), allow for great generality, these do not always lead to tractable forms of the transition probability. Indeed, the type of transition mechanism characterizing f -stationary Poisson-driven Markov processes leads to an effective Markov chain Monte Carlo (MCMC)-based estimation procedure.…”

“…An approach that enriches such thinning operators is due to Joe (1996) and Jørgensen and Song (1996), where discretetime Markov processes with invariant distributions in the convolution-closed infinitely divisible class are presented. Onedimensional stationary diffusion processes of the mean reverting type, with prescribed marginal distributions, are explored in Bibby, Skovgaard, and Sørensen (2005) by specifying a particular form of the diffusion coefficient modulating the process.…”

An introduction to Bent Jørgensen’s ideas

“…3.3.1 Latent Stochastic Processes Based Models Dispersion and ED models were used to study data containing clustered and dependent observations (Jørgensen et al, 1996a,b,c;Jørgensen and Song, 1997;Jørgensen and Tsao, 1999;Artes and Jørgensen, 2000;Botter et al, 2002;Ma and Jørgensen, 2007;Ma et al, 2009) in recent years. The common idea explored there is that the dependence in the data is modeled using a latent stochastic process, the observations being conditionally independent given the latent process.…”

“…A group of statisticians was quickly gathered by Bent Jørgensen at the UBC (e.g., Søren Lundbye-Christensen, as a recurrent visitor from Denmark, and Peter Song, among others). They worked in a new research line in which EDMs were used to represent a latent stochastic process governing the temporal development of a phenomena of interest, see Jørgensen et al (1996a,b,c); Jørgensen and Song (1997) and Section 3.3.1. In this period (around 1995-1997) Bent Jørgensen worked also with Rinaldo Artes (currently at Insper -Instituto de Ensino e Pesquisa, São Paulo) developing part of his thesis entitled "Extensions of generalized estimation equation theory to circular data and dispersion models", which was approved in 1997 at University of São Paulo, see also Artes and Jørgensen (2000).…”

Thinning-based models in the analysis of integer-valued time series: a review

“…The quasi-binomial thinning of Alzaid and Al-Osh (1993) is a special case of the random operator of Joe (1996) in which the distributions of Xt, Zt and Mt are generalized Poisson with parameters false(θ,λfalse), false(θ,false(1−αfalse)λfalse) and false(θ,αλfalse), respectively. Extensions of Joe's results for higher order INARMA models have been considered by Jørgensen and Song (1998) and Jung and Tremayne(2011b).…”