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The max-INAR(1) model for count processes


This paper proposes a discrete counterpart of the conventional max-autoregressive process of order one. It is based on the so-called binomial thinning operator and driven by a sequence of independent and identically distributed nonnegative integer-valued random variables with either regularly varying right tail or exponential-type right tail. Basic probabilistic and statistical properties of the process are discussed in detail, including the analysis of conditional moments, transition probabilities, the existence and uniqueness of a stationary distribution, and the relationship between the observations’ and innovations’ distribution. We also provide conditions on the marginal distribution of the process to ensure that the innovations’ distribution exists and is well defined. Several examples of families of distributions satisfying such conditions are presented, but also some counterexamples are analyzed. Furthermore, the analysis of its extremal behavior is also considered. In particular, we look at the limiting distribution of sample maxima and its corresponding extremal index.

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  • Al-Osh MA, Alzaid AA (1987) First order integer-valued autoregressive INAR(1) process. J Time Ser Anal 8:261–275

    MathSciNet  Article  Google Scholar 

  • Alpuim MT (1989) An extremal Markovian sequence. J Appl Probab 26:219–232

    MathSciNet  Article  Google Scholar 

  • Anderson CW (1970) Extreme value theory for a class of discrete distributions with applications to some stochastic processes. J Appl Probab 7:99–113

    MathSciNet  Article  Google Scholar 

  • Chernick M (1981) On strong mixing and Leadbetter’s D condition. J Appl Probab 18:764–769

    MathSciNet  Article  Google Scholar 

  • Christoph G, Schreiber K (2000) Scaled Sibuya distribution and discrete self-decomposability. Stat Probab Lett 48:181–187

    MathSciNet  Article  Google Scholar 

  • Cline DBH (1986) Convolution tails, product tails and domains of attraction. Probab Theory Relat Fields 72:529–557

    MathSciNet  Article  Google Scholar 

  • Daley D, Haslet J (1982) A thermal energy storage with controlled input. Adv Appl Probab 14:257–271

    MathSciNet  Article  Google Scholar 

  • Davis RA, Resnick SI (1989) Basic properties and prediction of Max-ARMA processes. Adv Appl Probab 21:781–803

    MathSciNet  Article  Google Scholar 

  • Feller W (1968) An introduction to probability theory and its applications, vol I. Wiley, New York

    MATH  Google Scholar 

  • Ferreira M, Canto e Castro L (2010) Modeling rare events through a pRARMAX process. J Stat Plann Inference 140:3552–3566

    Article  Google Scholar 

  • Ferreira M, Ferreira H (2013) Extremes of multivariate ARMAX processes. Test 22:606–627

    MathSciNet  Article  Google Scholar 

  • Grunwald G, Hyndman RJ, Tedesco L, Tweedie R (2000) Non-Gaussian conditional linear AR(1) models. Aust N Z J Stat 42:479–495

    MathSciNet  Article  Google Scholar 

  • Hall A (1996) Maximum term of a particular autoregressive sequence with discrete margins. Commun Stat Theory Methods 25:721–736

    MathSciNet  Article  Google Scholar 

  • Hall A (2001) Extremes of integer-valued moving averages models with regularly varying tails. Extremes 4:219–239

    MathSciNet  Article  Google Scholar 

  • Hall A (2003) Extremes of integer-valued moving averages models with exponential type tails. Extremes 4:361–379

    MathSciNet  Article  Google Scholar 

  • Hall A, Scotto MG (2006) Extremes of periodic integer-valued sequences with exponential type tails. Revstat Stat J 4:249–273

    MathSciNet  MATH  Google Scholar 

  • Heathcote C (1966) Corrections and comments of the paper A branching process allowing immigration. J R Stat Soc B 28:213–217

    MathSciNet  Google Scholar 

  • Leadbetter MR, Nandagopalan S (1989) On exceedance point processes for stationary sequences under mild oscillation restrictions. In: Hüsler J, Reiss RD (eds) Extreme value theory: proceedings, Oberwolfach 1987. Lecture notes in statistics, vol 51. Springer, Berlin, pp 69–80

    Chapter  Google Scholar 

  • Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer, Berlin

    Book  Google Scholar 

  • Littlejohn RP (1992) Discrete minification processes and reversibility. J Appl Probab 29:82–91

    MathSciNet  Article  Google Scholar 

  • McKenzie E (1985) Measuring serial dependence in categorical time series. Water Resour Bull 21:645–650

    Article  Google Scholar 

  • Naveau P, Zhang Z, Zhu B (2011) An extension of max autoregressive models. Stat Interface 4:253–266

    MathSciNet  Article  Google Scholar 

  • Rootzén H (1986) Extreme value theory for moving average processes. Ann Probab 14:612–652

    MathSciNet  Article  Google Scholar 

  • Schweer S, Weiß CH (2014) Compound poisson INAR (1) processes: stochastic properties and testing for overdispersion. Comput Stat Data Anal 77:267–284

    MathSciNet  Article  Google Scholar 

  • Scotto MG, Weiß CH, Gouveia S (2015) Thinning-based models in the analysis of integer-valued time series: a review. Stat Model 15:590–618

    MathSciNet  Article  Google Scholar 

  • Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York

    Book  Google Scholar 

  • Steutel FW, van Harn K (1979) Discrete analogues of self-decomposability and stability. Ann Probab 7:893–899

    MathSciNet  Article  Google Scholar 

  • Steutel FW, van Harn K (2004) Infinite divisibility of probability distributions on the real line. Marcel Dekker, Inc, New York

    MATH  Google Scholar 

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The authors thank the referees for carefully reading the article and for their comments, which greatly improved the article.

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Correspondence to Manuel G. Scotto.

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This research was supported by the German Academic Exchange Service (DAAD) and the Fundação para a Ciência e a Tecnologia (FCT), under the program “Ações Integradas Luso-Alemãs” and the Grants 57212119 and A-38/16. Manuel Scotto also acknowledges the Project UID/Multi/04621/2013. S. Gouveia acknowledges the postdoctoral Grant by FCT (ref. SFRH/BPD/87037/2012). This work was also partially supported by the Portuguese FCT, with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020—within IEETA/UA Project UID/CEC/00127/2013 (Instituto de Engenharia Electrónica e Informática de Aveiro, IEETA/UA, Aveiro) and CIDMA/UA Project UID/MAT/04106/2013 (Centro de Investigação e Desenvolvimento em Matemática e Aplicações, CIDMA/UA, Aveiro).


A Conditional variance for geometric innovations

Let us pick up the situation of Example 1. For the second factorial moment, we compute

$$\begin{aligned} \sum \limits _{k=0}^{m}\ k_{(2)}\,P(\epsilon _t=k)= & {} q(1-q)^2\, \sum \limits _{k=2}^{m} k_{(2)}\,(1-q)^{k-2}\\= & {} 2\,\frac{(1-q)^2}{q^2}-\frac{(1-q)^{m+1}}{q^2}\,\big (m_{(2)}\,q^2+2\,m\,q+2(1-q)\big ). \end{aligned}$$


$$\begin{aligned} E\big [\max {\{m,\ \epsilon _t\}}_{(2)}\big ] =m_{(2)}\ +\ 2\,\frac{(1-q)^{m+1}}{q^2}\,(m\,q+1-q). \end{aligned}$$

Therefore, the conditional 2nd factorial moment of the geometric max-INAR(1) model equals

$$\begin{aligned} \begin{array}{rl} E\big [(X_t)_{(2)}\ |\ X_{t-1}&{} =l\big ] l_{(2)}\,\alpha ^2\ +\ 2\,\frac{(1-q)^2}{q}\,E_{\hbox {Bin}(l,\alpha )} \big [Y\,(1-q)^{Y-1}\big ]\\ &{}\quad +\ 2\,\frac{(1-q)^{2}}{q^2}\,(1-\alpha \,q)^l, \end{array} \end{aligned}$$

where the last term stems from the pgf of the \(\hbox {Bin}(l,\alpha )\)-distribution. In general, we have

$$\begin{aligned}&{\hbox {pgf}}_{\hbox {Bin}(l,\alpha )}(z) =\ \sum _{m=0}^l\ \left( {\begin{array}{c}l\\ m\end{array}}\right) \,\alpha ^m\,(1-\alpha )^{l-m}\cdot z^m. \end{aligned}$$

Differentiating both sides w.r.t. z, it follows that

$$\begin{aligned} l\,\alpha \,(1-\alpha +\alpha \,z)^{l-1}\ =\ \sum _{m=1}^l\ \left( {\begin{array}{c}l\\ m\end{array}}\right) \,\alpha ^m\,(1-\alpha )^{l-m}\cdot m\,z^{m-1}. \end{aligned}$$

So we continue

$$\begin{aligned} \begin{array}{rl} E\big [(X_t)_{(2)}\ |\ X_{t-1}=l\big ] \ =&l_{(2)}\,\alpha ^2\ +\ 2\,\frac{(1-q)^2}{q}\,l\,\alpha \,(1-\alpha \,q)^{l-1} \ +\ 2\,\frac{(1-q)^{2}}{q^2}\,(1-\alpha \,q)^l. \end{array} \end{aligned}$$

The conditional variance follows as

$$\begin{aligned} \begin{array}{rl} V[X_t\ |\ X_{t-1}=l] \ =&{} l_{(2)}\,\alpha ^2\ +\ 2\,\frac{(1-q)^2}{q}\,l\,\alpha \,(1-\alpha \,q)^{l-1}\ +\ 2\,\frac{(1-q)^{2}}{q^2}\,(1-\alpha \,q)^l\\ &{} +\ \Big (\frac{1-q}{q}\,(1-\alpha \,q)^l\, +\, l\,\alpha \Big )\ -\ \Big (\frac{1-q}{q}\,(1-\alpha \,q)^l\, +\, l\,\alpha \Big )^2\\ =&{}\alpha (1-\alpha )\ -\ \frac{(1-q)^2}{q^2}\,(1-\alpha \,q)^{2l}\\ &{} +\ \frac{1-q}{q}\,(1{-}\alpha \,q)^{l-1}\, \Big ( {-}2\,l\,\alpha (1{-}\alpha )q {+} 2\,\frac{1-q}{q}-2\alpha + 1+\alpha \,q \Big ). \end{array} \end{aligned}$$

B Long-range and local dependence conditions

This appendix includes the definition of Leadbetter’s \(D(u_n)\) condition and the local dependence conditions \(D^{(1)}(u_n)\) and \(D^{(2)}(u_n)\).

Definition 1

The condition \(D(u_n)\) is said to hold for a stationary sequence \((X_t)\) if for any p, \(p'\) and n such that \(1\le i_1<\cdots<i_p<j_1<\cdots <{j_{p'}}\le n,\,(j_1-i_p\ge l)\), we have

$$\begin{aligned} \left| P(\max _{i\in A\cup B}X_i\le u_n)-P(\max _{i\in A}X_i\le u_n)P(\max _{i\in B}X_i\le u_n)\right| \le \gamma _{n,l}, \end{aligned}$$

where \(A=\{i_1,\dots ,i_p\}\), \(B=\{j_1,\dots ,i_{p'}\}\) and \(\gamma _{n,l}\rightarrow 0\) for some sequence \(l=l_n=o(n)\).

Definition 2

The condition \(D^{(1)}(u_n)\) is said to hold for a stationary sequence \((X_t)\) if there exists a sequence of integer values \((s_n)\) with \(s_n\rightarrow \infty \), such that

$$\begin{aligned} \limsup _{n\rightarrow \infty } n \sum _{j=2}^{[n/s_n]}P(X_1> u_n, X_j> u_n)=0. \end{aligned}$$

Definition 3

The process \((X_t)\) verifies condition \(D^{(2)}(u_n)\) if there exist sequences of integers \((s_n)\) and \((l_n)\) such that \(s_n\rightarrow \infty \), \(s_n\gamma _{n,l_n}\rightarrow 0\), \(\frac{s_nl_n}{n}\rightarrow 0\), as \(n\rightarrow \infty \), and

$$\begin{aligned} \lim _{n\rightarrow \infty }nP(X_1>u_n\ge X_2, M_{3, [n/s_n]}>u_n)=0, \end{aligned}$$

where \(M_{i,j}:=\max {\{X_i,\dots ,X_j\}}\), for \(i\le j\) and \(-\infty \), otherwise. Note that the above condition is implied by the condition

$$\begin{aligned} \lim _{n\rightarrow \infty }n\sum _{j=3}^{[n/s_n]}P(X_1>u_n\ge X_{2}, X_j>u_n)=0. \end{aligned}$$

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Scotto, M.G., Weiß, C.H., Möller, T.A. et al. The max-INAR(1) model for count processes. TEST 27, 850–870 (2018).

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  • Time series of counts
  • Thinning operator
  • Autoregressive processes
  • Extremal index

Mathematics Subject Classification

  • 62M10
  • 60G70