The max-INAR(1) model for count processes

Abstract

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.

Appendices

A Conditional variance for geometric innovations

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

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

So

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

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

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