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First-order rounded integer-valued autoregressive (RINAR(l)) process. (English) Zbl 1224.62060

The integer-valued autoregressive (INAR) processes, introduced by E. McKenzie [Some simple models for discrete variate time series. Water Resources Bull. 21, 645-650 (1985)] and M.A. Al-Osh and A.A. Alzaid [J. Time Ser. Anal. 8, 261–275 (1987; Zbl 0617.62096)], are based on the thinning operator $\circ$ defined as $a\circ X={\sum }_{k=1}^{X}{\xi }_{k}$, where $X$ is a non-negative integer-valued random variable, $a\in \left[0,1\right]$, and $\left\{{\xi }_{k}\right\}$ are i.i.d. Bernoulli random variables, independent of $X$ and satisfying $P\left\{{\xi }_{k}=1\right\}=a=1-P\left\{{\xi }_{k}=0\right\}$. A sequence $\left\{{X}_{t},\phantom{\rule{4pt}{0ex}}t\in ℤ\right\}$ is said to be an INAR(p) process if it admits the representation

${X}_{t}=\sum _{j=1}^{p}{a}_{j}\circ {X}_{t-j}+{\epsilon }_{t},\phantom{\rule{1.em}{0ex}}{a}_{j}\circ {X}_{t-j}=\sum _{k=1}^{{X}_{t-j}}{\xi }_{jt}\left(t\right),\phantom{\rule{1.em}{0ex}}t\in ℤ,$

where $\left\{{\epsilon }_{t}\right\}$ are non-negative i.i.d. integer-valued random variables independent of all the counting series ${\xi }_{jt}\left(t\right),\phantom{\rule{4pt}{0ex}}t\in ℤ,\phantom{\rule{4pt}{0ex}}j,k\in ℕ$, $E{\epsilon }_{t}=b$, ${a}_{j}\in \left[0,1\right],\phantom{\rule{4pt}{0ex}}j=1,\cdots ,p$. For a review of discrete variate time series models see E. McKenzie [D.N. Shanbhag et al. (eds.), Stochastic processes: Modelling and simulation. Amsterdam: North-Holland. Handb. Stat. 21, 573–606 (2003; Zbl 1064.62560)]. The class of INAR models has many limitations. Their innovation structure is complex, depending not only on the noise process $\left\{{\epsilon }_{t}\right\}$, but also on the counting variables ${\xi }_{jt}\left(t\right)$.

In this work the authors introduce the rounded integer-valued autoregression RINAR(p) process that admits the representation

${X}_{t}=〈\sum _{j=1}^{p}{\alpha }_{j}{X}_{t-j}+\lambda 〉+{\epsilon }_{t},\phantom{\rule{4pt}{0ex}}t\in ℕ,$

where $〈·〉$ represents the rounding operator to the nearest integer, $\left\{{\epsilon }_{t}\right\}$ are centred i.i.d. integer-valued random variables, and $\lambda$ and $\left\{{\alpha }_{j}\right\}$ are real parameters. The RINAR(p) model is a direct and natural extension on $ℤ$ of the real AR(p) model. In fact, the rounding operator is considered as a censoring function on the real AR(p) model. RINAR(p) has many advantages compared with the INAR models. Its innovation structure is simple, generated only by the noise $\left\{{\epsilon }_{t}\right\}$, its stationarity is ensured under general conditions on the parameters $\left\{{\alpha }_{j}\right\}$ which are similar to that for AR(p). The RINAR(p) model can be used to analyse time series with negative values, a situation not covered by INAR models.

In this article, the authors deal with the simplest RINAR(l) models. Conditions are described ensuring the stationarity and ergodicity of the models. The least squares estimator is introduced for estimation of the model parameters. It is proved that the proposed estimator is consistent under suitable conditions. A simulation experiment is presented to test the performance of the estimator. An analysis of two real data sets is performed with a RINAR(l) model. The first series shows some negative autocorrelations, while the second series has itself some negative values so that none of them can be fitted with the classical INAR(l) model.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 62F12 Asymptotic properties of parametric estimators 65G50 Roundoff error 65C60 Computational problems in statistics
EDDIE