Integer-valued auto-regressive (INAR) processes have been introduced to model non-negative integer-valued phenomena that evolve over time. The distribution of an INAR(p) process is essentially described by two parameters: a vector of auto-regression coefficients and a probability distribution on the non-negative integers, called an immigration or innovation distribution. Traditionally, parametric models are considered where the innovation distribution is assumed to belong to a parametric family. The paper instead considers a more realistic semiparametric INAR(p) model where there are essentially no restrictions on the innovation distribution. We provide an (semiparametrically) efficient estimator of both the auto-regression parameters and the innovation distribution.

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Integer-valued autoregressive (INAR) processes have been introduced to model nonnegative integervalued phenomena that evolve in time. The distribution of an INAR(p) process is determined by two parameters: a vector of survival probabilities and a probability distribution on the nonnegative integers, called an immigration or innovation distribution. This paper provides an efficient estimator of the parameters, and in particular, shows that the INAR(p) model has the Local Asymptotic Normality property.

This paper considers non-negative integer-valued autoregressive processes where the autoregression parameter is close to unity. We consider the asymptotics of this 'near unit root' situation. The local asymptotic structure of the likelihood ratios of the model is obtained, showing that the limit experiment is Poissonian. To illustrate the statistical consequences we discuss efficient estimation of the autoregression parameter and efficient testing for a unit root.