Abstract
A Poisson distribution is commonly used as the innovation distribution for integer-valued autoregressive models, but its mean is equal to its variance, which limits flexibility, so a flexible, one-parameter, infinitely divisible Bell distribution may be a good alternative. In addition, for a parameter with a small value, the Bell distribution approaches the Poisson distribution. In this paper, we introduce a new first-order, non-negative, integer-valued autoregressive model with Bell innovations based on the binomial thinning operator. Compared with other models, the new model is not only simple but also particularly suitable for time series of counts exhibiting overdispersion. Some properties of the model are established here, such as the mean, variance, joint distribution functions, and multi-step-ahead conditional measures. Conditional least squares, Yule–Walker, and conditional maximum likelihood are used for estimating the parameters. Some simulation results are presented to access these estimates’ performances. Real data examples are provided.
Highlights
In recent years, studying count time series has attracted a lot of attention in different fields, such as finance, medical science, and insurance
We found that the Akaike’s information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), and Hannan–Quinn information criterion (HQIC)
A new INAR(1) model with Bell innovations based on the binomial thinning operator was introduced in this paper
Summary
In recent years, studying count time series has attracted a lot of attention in different fields, such as finance, medical science, and insurance. Many data examples are overdispersed (variance is greater than mean) relative to the Poisson distribution. For this reason, the INAR(1) model with Poisson innovations is not always suitable for modeling integer-valued time series. Jazi et al (2012) [10] proposed a modification of the INAR(1) model with geometric innovations (G-INAR(1)) for modeling overdispersed count data. By comparing the results of different information criteria, it can be seen that the BL-INAR(1) model is competitive when modeling the overdispersed integer-valued time series data, which shows that the proposed BL-INAR(1) model is meaningful; see Section 5 for more details. Later we introduce the BL-INAR(1) model and derive some basic properties of it
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