Poisson limit of an inhomogeneous nearly critical INAR(1) model

We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of AR(1) process with random-coefficient $a \in [0,1)$ when $N$ and time scale $n$ increase at different rate. Assuming that $a$ has a density, regularly varying at $a = 1$ with exponent $-1 < \beta < 1$, different joint limits of normalized aggregated partial sums are shown to exist when $N^{1/(1+\beta)}/n$ tends to (i) $\infty$, (ii) 0, (iii) $0 < \mu < \infty$. The limit process arising und...

In this paper the integer-valued autoregressive model of order one, contaminated with additive or innovational outliers is studied in some detail. Moreover, parameter estimation is also addressed. Supposing that the time points of the outliers are known but their sizes are unknown, we prove that the Conditional Least Squares (CLS) estimators of the offspring and innovation means are strongly consistent. In contrast, however, the CLS estimators of the outliers' sizes are not s...

Outlying observations are commonly encountered in the analysis of time series. In this paper the problem of detecting additive outliers in integer-valued time series is considered. We show how Gibbs sampling can be used to detect outlying observations in INAR(1) processes. The methodology proposed is illustrated using examples as well as an observed data set.

We derive mixing properties for a broad class of Poisson count time series satisfying a certain contraction condition. Using specific coupling techniques, we prove absolute regularity at a geometric rate not only for stationary Poisson-GARCH processes but also for models with an explosive trend. We provide easily verifiable sufficient conditions for absolute regularity for a variety of models including classical (log-)linear models. Finally, we illustrate the practical use of...

In this note, we proved that weak limits, of sums of independent positive identically distributed random variables which are re-normalized by a non-linear shrinking transform $\max(0, x-r)$, are either degenerate or (some) compound Poisson distributions.

In Fernandez-Fontelo et al (Statis. Med. 2016, DOI 10.1002/sim.7026) hidden integer-valued autoregressive (INAR) processes are used to estimate reporting probabilities for various diseases. In this comment it is demonstrated that the Poisson INAR(1) model with time-homogeneous underreporting can be expressed equivalently as a completely observed INAR(inf) model with a geometric lag structure. This implies that estimated reporting probabilities depend on the assumed lag struct...

INteger Auto-Regressive (INAR) processes are usually defined by specifying the innovations and the operator, which often leads to difficulties in deriving marginal properties of the process. In many practical situations, a major modeling limitation is that it is difficult to justify the choice of the operator. To overcome these drawbacks, we propose a new flexible approach to build an INAR model: we pre-specify the marginal and innovation distributions. Hence, the operator is...

We extend classical results about the convergence of nearly unstable AR(p) processes to the infinite order case. To do so, we proceed as in recent works about Hawkes processes by using limit theorems for some well chosen geometric sums. We prove that when the coefficients sequence has a light tail, infinite order nearly unstable autoregressive processes behave as Ornstein-Uhlenbeck models. However, in the heavy tail case, we show that fractional diffusions arise as limiting l...

In earlier stages in the introduction to asymptotic methods in probability theory, the weak convergence of sequences $(X_n)_{n\geq 1}$ of Binomial of random variables (\textit{rv}'s) to a Poisson law is classical and easy-to prove. A version of such a result concerning sequences $(Y_n)_{n\geq 1}$ of negative binomial \textit{rv}'s also exists. In both cases, $X_n$ and $Y_n-n$ are by-row sums $S_n[X]$ and $S_n[Y]$ of arrays of Bernoulli \textit{rv}'s and corrected geometric \t...

Contemporaneous aggregation of individual AR(1) random processes might lead to different properties of the limit aggregated time series, in particular, long memory (Granger, 1980). We provide a new characterization of the series of autoregressive coefficients, which is defined from the Wold representation of the limit of the aggregate stochastic process, in the presence of long-memory features. Especially the infinite autoregressive stochastic process defined by the almost su...