Modelling Count Data Time Series with Markov Processes Based on Binomial Thinning

Abstract

We obtain new models and results for count data time series based on binomial thinning. Count data time series may have non-stationarity from trends or covariates, so we propose an extension of stationary time series based on binomial thinning such that the univariate marginal distributions are always in the same parametric family, such as negative binomial. We propose a recursive algorithm to calculate the probability mass functions for the innovation random variable associated with binomial thinning. This simplifies numerical calculations and estimation for the classes of time series models that we consider. An application with real data is used to illustrate the models. Copyright 2006 The Authors Journal compilation 2006 Blackwell Publishing Ltd.

Thinning operations for modeling time series of counts—a survey

“…Two types of INAR(p) models were proposed by Alzaid and Al-Osh (1990) and Du and Li (1991); a full INARMA(p, q) model was put forth by Dion et al (1995). An alternative AR(p)-like approach, designed for DSD marginals, is due to Zhu and Joe (2006) and Weiß (2008). Properties of the INAR(p) model of Du and Li (1991), but with generalized instead of binomial thinning, were discussed by, among others, Latour (1998), da Silva andOliveira (2005) and da Silva and da Silva (2006).…”

A Poisson INAR(1) process with a seasonal structure

“…This series was previously studied by Freeland [15] and Zhu and Joe. [11] In the former study, covariates were used to account for the seasonality feature. In the same direction, the latter also considered covariates functions to explain seasonality in counting models with Poisson and negative binomial marginals to adjust the series.…”

“…These models are not stationary. Zhu and Joe [11] also fitted the series using INAR(1) processes with Poisson and negative binomial innovations without consideration of seasonality. As was expected, the models with covariates, that is, the non-stationary processes, produced smaller Akaike information criterion (AIC) values, however with larger parametric dimensions.…”

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

“…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. Applications of time series with discrete marginal distributions based on thinning operators have been found in the following areas: medical science (Latour 1995), compensation claims (Freeland and McCabe 2004;Zhu and Joe 2006) and academic research with regards to abstract review counts (Zhu and Joe 2010) as well as crime count data (Ristic et al 2009). Jung and Tremayne (2011) have compared and contrast a variety of time series models for counts using two very different data sets.…”