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In this paper, a first-order integer-valued autoregressive (INAR(1)) model with zero-and-one inflated Poisson–Lindley distributed innovations is presented. It is shown that the model represents a mixture of zero-and-one inflated geometric INAR(1) process and zero-and-one inflated negative binomial INAR(1) process. Moreover, some basic and conditional properties of this model are obtained. The distribution of the runs (the lengths of zeros and ones) is investigated. Some estimation methods are used to estimate the unknown parameters of the model and the asymptotic properties of the estimators are obtained. Using the conditional maximum likelihood estimation method, the model is fitted to a set of practical data, and the model is validated by some goodness-of-fit criteria.

Zero-and-one inflated Poisson–Lindley INAR(1) process for modelling count time series with extra zeros and ones