Autoregressive moving-average processes with negative-binomial and geometric marginal distributions

Abstract

Some simple models are described which may be used for the modelling or generation of sequences of dependent discrete random variates with negative binomial and geometric univariate marginal distributions. The models are developed as analogues of well-known continuous variate models for gamma and negative exponential variates. The analogy arises naturally from a consideration of self-decomposability for discrete random variables. An alternative derivation is also given wherein both the continuous and the discrete variate processes arise simultaneously as measures on a process of overlapping intervals. The former is the process of interval lengths and the latter is a process of counts on these intervals.

Thinning operations for modeling time series of counts—a survey

“…This special type of random coefficient thinning was also proposed earlier by McKenzie (1985McKenzie ( , 1986. It was applied by Jung et al (2005) to generate count data processes exhibiting overdispersion, also see the subsequent discussion.…”

“…Here, φ has to follow an appropriate beta distribution. Such beta-binomial thinning operations were considered by McKenzie (1985McKenzie ( , 1986, Joe (1996) and Jung et al (2005). It is clear by definition that the conditional distribution of φ • X for a given X = x is the beta-binomial distribution BB(x; α, β) …”

On the Stationary Version of the Generalized Hyperbolic ARCH Model

“…The general exposition of these models resulted in an exponential autoregressive-moving average ( EARMA(p, q) ) model presented in Lawrance and Lewis (1980). Other models with marginal distributions belonging to other families are found in Gaver and Lewis (1980) and Lawrance (1982) for mixed exponential distributions and for gamma distributions; McKenzie (1986McKenzie ( , 1988 for negative binomial and Poisson distributions; Joe (1996) for infinitely divisible convolution-closed distributions; and Jørgensen and Song (1998) for convolution-closed exponential dispersion distributions. Some other accounts are found in Lawrance (1991) and Jørgensen and Song (1998).…”

Risk models based on time series for count random variables

“…These models are based on appropriate thinning operations which replace the scalar multiplications in the Gaussian ARMA framework of time series consisting of continuous data (see e.g. Al-Osh and Alzaid (1987,1988), McKenzie (1986McKenzie ( , 1988McKenzie ( , 2003 and Joe (1997)). …”