A time series model based on dependent zero inflated counting series
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A time series model based on dependent zero inflated counting series Nisreen Shamma1 · Mehrnaz Mohammadpour1 · Masoumeh Shirozhan1 Received: 10 June 2019 / Accepted: 21 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we introduce a new generalized negative binomial thinning operator with dependent counting series. Some properties of the thinning operator are derived. A new stationary integer-valued autoregressive model based on the thinning operator is constructed. In addition various properties of the process are determined, unknown parameters are estimated by several methods and the behavior of the estimators is described through the numerical results. Also, the model is applied on a real data set and compared to some relevant INAR(1) models. Keywords Dependent thinning operator · INAR model · Modified conditional least square method · Overdispersion
1 Introduction Integer-valued autoregressive (INAR) processes with dependent counting series are an adaptation of INAR processes to dependent counting variables by considering the dependence on current population via its thinning operator. These processes are suitable for modeling the series affecting to each others or interacting among themselves such as the spread of parasitic diseases, the commission of vagrancy offences and survival or collapse of some companies in economy. The INAR process with dependent counting series has been first proposed by Risti´c et al. (2013) while the most common studies on modeling count time series are based on the thinning operator with independent counting series. The INAR(1) process was introduced by Al-Osh and Alzaid (1987). Among independent counting INAR(1) models, we cite the Poisson INAR(1) model (Al-Osh and Alzaid 1987), geometric INAR(1) model (Alzaid and Al-Osh 1988), generalized Poisson INAR(1) model (Alzaid and Al-Osh 1993), negative binomial INAR(1) model
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Mehrnaz Mohammadpour [email protected] Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
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(Al-Osh and Aly 1992; Risti´c et al. 2009), geometric INAR(1) model based on inflatedBernoulli counting variables (Borges et al. 2016), geometric INAR(1) model based on inflated-geometric counting variables (Borges et al. 2017), geometric INAR(1) model based on deflated and inflated counting variables (Bourguignon et al. 2018) and zeroinflated geometric INAR(1) model with random coefficient (Bakouch et al. 2018). A significant development of the independent counting series was made by Mileti´c Ili´c et al. (2018) as all counting variables are not independent. Risti´c et al. (2013) offered dependent Bernoulli thinning operators, Nasti´c et al. (2017) modify the definition of dependent Bernoulli thinning operators to make it more simple and flexible to work, Mileti´c Ili´c (2016) generalized thinning operator based on dependent Bernoulli counting variables and Shirozhan et al. (2019) mixed the dependent thinning operator with Pegram operator. In this paper, we study an INA
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