Page 119 - Textos de Matemática Vol. 47

MODELLING TIME SERIES OF COUNTS: AN INAR APPROACH 109
The intuition behind the operator 0⇧0, as described by that author is the fol- lowing.
Let F✓ denote a convolution-closed infinitely divisible parametric family such that F✓1 ⇤F✓2 = F✓1+✓2 , with 0⇤0 denoting the convolution operator. Let Y1, Y2 be independent random variables each with distribution F(1 ↵)✓ (pmf f(1 ↵)✓(·)) and Y12 be a another random variable independent from Y1, Y2 with distribution F↵✓ . The distribution of Y12 given Y12 + Y1 = y is denoted by G↵✓,(1 ↵)✓,y and its pmf by g(·|y). [25] writes the joint distribution of (Xt,Xt 1) as being the sameasthatof(Y12+Y2,Y12+Y1),inwhichcaseY12 representsacommonlatent or unobserved component of the pair (Xt, Xt 1) that carries the dependence of the observations between two consecutive time periods and Yi, i = 1, 2 represent the arrivals in the model.
[25] shows that the processes defined in (2.1) are Markov order 1, time reversible and stationary with non-negative serial dependence, ⇢k = ↵k. The transition probabilities are given by
P(Xt = k|Xt 1 = `) = =
miXn{k,`} j=0
miXn{k,`} j=0
g(j|`)P(✏t = k j) g(j|`)f(1 ↵)✓(k j).
(2.2)
Moreover, the INAR(1) model is a member of the class of conditional linear first order autoregressive, CLAR(1), models introduced by [20].
2.1. The Poisson INAR(1). Consider the case where F✓ is Poisson ✓ = /(1 ↵). Setting G↵✓,(1 ↵)✓,x as Binomial(x, ↵), leads to the most common thinning operator which is the binomial thinning, denoted by 0 0 and originally introduced by [46] to extend the notions of self-decomposability (DSD) and stability to integer-valued time series.
Definition 2.2. Let X be a non-negative integer-valued random variable. Then, for any ↵ 2 [0, 1] define the binomial thinning operator as
X i=1
where {Yi}i is a sequence of independent and identically distributed Bernoulli random variables with P(Yi = 1) = ↵, called the counting series of ↵ X, which is also independent of X.
For properties of the binomial thinning operation see [47, 51, 43, 44].
↵ X :=
Yi, (2.3)