Asymptotic inference for nearly unstable INAR(1) models

Abstract

The first-order integer-valued autoregressive (INAR(1)) process is investigated, where the autoregressive coefficient is close to one. It is shown that the limiting distribution of the conditional least-squares estimator for this coefficient is normal and, in contrast to the familiar AR(1) process, the rate of convergence is n 3/2 . Finally, the nearly critical Galton-Watson process with unobservable immigration is discussed.

Fluctuation limit of branching processes with immigration and estimation of the means

“…The norming factor for the offspring mean is n 3/2 , in contrast to both the subcritical case, in which it is n 1/2 , and the nearly critical case with positive limiting offspring variance, in which it is n. We remark that the results of the present paper are generalizations of those of [7] and [8], where a Bernoulli offspring distribution was assumed.…”

“…This special case was investigated in [7] and [8]. If the offspring distributions are geometric distributions with parameter p n = 1 − an −1 , where a ≥ 0, i.e.…”

“…By applying the continuous-mapping theorem and using Slutsky's argument, in the nearly critical model of Theorem 2.2 we can derive the asymptotic behaviour of the estimators m n and ( m n , λ n ) in exactly the same way as was done in the case of a Bernoulli offspring distribution in [7] and [8]. …”

Efficient Probabilistic Forecasts for Counts

“…It was further investigated by, among others, Du and Li (1991), Dion et al. (1995), Latour (1998), Ispany et al. (2003), Freeland and McCabe (2004), Jung et al.…”

Asymptotic Behavior of Conditional Least Squares Estimators for Unstable Integer‐valued Autoregressive Models of Order 2

“…Du and Li (), in Theorem , proved asymptotic normality of the conditional least squares (CLS) estimator of the autoregressive parameters for stable INAR( p ) models; see also Proposition 6.1 of Latour (); Brännäs and Hellström () considered generalized method of moment estimation. Ispány et al (), () derived asymptotic inference for nearly unstable INAR(1) models, which has been refined by Drost et al () later. In Ispány et al (), the mean of the innovations was supposed to be known, whereas in Ispány et al (), both the autoregressive parameter and the mean of the innovations have been estimated jointly.…”