A generalised NGINAR(1) process with inflated‐parameter geometric counting series

Abstract

Summary In this paper we propose a new stationary first‐order non‐negative integer valued autoregressive process with geometric marginals based on a generalised version of the negative binomial thinning operator. In this manner we obtain another process that we refer to as a generalised stationary integer‐valued autoregressive process of the first order with geometric marginals. This new process will enable one to tackle the problem of overdispersion inherent in the analysis of integer‐valued time series data, and contains the new geometric process as a particular case. In addition various properties of the new process, such as conditional distribution, autocorrelation structure and innovation structure, are derived. We discuss conditional maximum likelihood estimation of the model parameters. We evaluate the performance of the conditional maximum likelihood estimators by a Monte Carlo study. The proposed process is fitted to time series of number of weekly sales (economics) and weekly number of syphilis cases (medicine) illustrating its capabilities in challenging cases of highly overdispersed count data.

A new approach to model the counts of earthquakes: INARPQX(1) process

“…After the important researches of McKenzie [17,18] and Al-Osh and Alzaid [2], the researches have focussed on the distribution of an innovation process of INAR (1) to develop new models for over-dispersed or under-dispersed time series of counts. In what follows, we list some recent contributions on overdispersed INAR(1) processes: INAR(1) process with geometric innovations (INARG(1)) by Jazi et al [13], INAR (1) with Poisson-Lindley innovations (INARPL(1)) by Lívio et al [16], compound Poisson INAR(1) by Schweer and Weiß [23], processes INAR(1) process with Katz family innovations by Kim and Lee [14], INAR(1) process with generalized Poisson and double Poisson innovations by Bourguignon et al [6] , INAR(1) process with geometric marginals by Borges et al [5] and INAR(1) process with Skellam innovations by Andersson and Karlis [4], INAR(1) process with a new Poisson-weighted exponential innovations by Altun [3].…”

Fractional approaches for the distribution of innovation sequence of INAR(1) processes

“…where ρ ∈ [0, 1) and µ > 0. The inflated-parameter geometric distribution is well-known as ρ-geometric distribution (see Kolev et al , 2000) and considered in a recent paper by Borges et al (2017) to formulate a new thinning operator. The corresponding pgf is given by…”

“…Nastić et al (2016b) introduced an r-states random environment non-stationary INAR(1) which, by its different values, represents the marginals selection mechanism from a family of different geometric distributions. Borges et al (2016) and Borges et al (2017) introduced geometric first-order integer-valued autoregressive processes with geometric marginal distribution based on ρ-binomial thinning operator and ρ-geometric thinning operator, respectively.…”

Extended Poisson INAR(1) processes with equidispersion, underdispersion and overdispersion

“…As a first example, we consider the data set consisting of the weekly number of syphilis cases in the United States from 2007 to 2010 in Mid-Atlantic states given in tsinteger package available for download at data(syphillis). The data consist of 209 observations, and they were already analyzed by Borges et al (2017).…”