Page 120 - Textos de Matemática Vol. 47

110 M. E. SILVA
The binomial thinning based INAR(1) model was originally proposed by [2] and [32]. The conditional distribution of Xt given Xt 1, fXt|Xt 1(xt|xt 1) = p(Xt|Xt 1) is now the convolution of the two components, binomial and Pois- son, as follows:
i X i e Xt i
↵ (1 ↵) t 1 , (2.4)
= exp ⇢ [2 ↵ (1 ↵)(s1 +s2) ↵s1s2] (2.5) 1 ↵
is symmetric in its arguments s1 and s2 and therefore the process is time- reversible. The PoINAR(1) may be interpreted as an infinite server queue. The service time is geometric with parameter 1 ↵ and the arrival process is Poisson with mean . A fundamental result in queueing theory, Little Flow’s equation, states that the expected length of the queue is equal to the arrival rate times the expected waiting time. It is thus possible to compute the expected number of time units a newly arrival stays in the system and which is given by: 1/(1 ↵).
3. Parameter Estimation and Diagnostic Tools
3.1. Parameter estimation. This section considers the estimation of the pa- rameters in the INAR(1) models discussed previously. Estimation can be car- ried out in several ways leading to the following broad categories of estimators: moment based estimators (MM), regression based or conditional least squares (CLS) estimators and likelihood based (ML) estimators. All these approaches have been considered in detail in the literature for the Poisson model. Addition- ally Bayesian methodology has been considered by [36] and [45]. However, the most common approach for the estimation of the INAR(1) model is maximum likelihood method.
Let x = (x1, . . . , xn) represent the observed time series and ✓ the s⇥1 vector of model parameters to be estimated. Since the INAR(1) model (2.1) is a first order stationary Markov chain, the likelihood function is written as
Yn t=2
Mt✓ ◆ X Xt 1
p(Xt|Xt 1)=
where Mt = min (Xt 1, Xt). The bivariate pgf of the PoINAR(1) process, given
i=0 i
(Xt i)!
by [4]
PXt,Xt 1 (s1, s2) =PXt 1 (s1(1 ↵ ↵s2))P✏t (s2)
Ln(✓|x) = P(X1 = x1)
fXt|Xt 1 (xt|xt 1), (3.1)