Page 123 - Contributed Paper Session (CPS) - Volume 2

CPS1458 KHOO W.C et al.
complex matrices. Zhu and Joe (2006) considered an alternative representative
of INAR(p) model by probability mass function, specifically discussing the case
of = 2. Then, Weiß (2008) extended the approach of Zhu and Joe (2006) to
the pth -order model by using probability generating function. A general
result on autocorrelation structure was also presented. If is a stationary

combined INAR(p) process, abbreviated by CINAR, then its probability
p
generating function (pgf) satisfies G X (z ) =  i= 1  i G X 1 ( −  + z  ) G  (z ) .
However, the CINAR(p) model only accommodates the DSD family. Unlike the
proposed MPT(p) model, it covers wider range of marginal distributions
including non DSD family such as Binomial distribution. Recent discussion in
Ristic and Nastic (2012) provided a different interpretation of the mixed
INAR(p) model. In this work a study of geometric mixed integer-valued
autoregressive INAR models was carried out. They considered the
combination of Bernoulli and Negative Binomial random variables for the
thinned counting time series. Such models involved intractable expressions
which are difficult to apply in practice. It is shown that the proposed MPT(p)
possesses great flexibility and has a simpler expression which is beneficial for
model interpretation. The construction of the MPT(p) model and its properties
will be discussed in the following Section.

3. Mixture of Pegram and thinning pth-order integer-valued
autoregressive (MPT(p)) process:
The construction of the pth -order integer-valued autoregressive model is
simple and is a natural extension of Equation (1). The mixture of Pegram-
INAR(p) model is defined as follows. Let be a discrete-valued stochastic

process and be an i.i.d process with range . Let ∈ (0,1) and be the

0

mixing weights where ∈ (0,1), = 1, … , , ∑ ∈ (0,1). For every =

=1
0, ±1, ±2, …. The process of ( ) is defined by

= ( , ∘ −1 ) ∗ ( , ∘ −2 ) ∗ … ∗ ( , ∘ − )

1

2

∗ (1 − − ⋯ − , ) (5)

1

The time index below the thinning operation indicates that the
corresponding thinning is used to defined the discrete-valued stochastic
process , and the notation * indicates Pegram's mixing operator. Equation

(5) is called mixture of Pegram and thinning of pth-order processes (MPT(p))
process if it satisfies the following assumptions:
(i) is independent with thinning at time (∘ ) , ∘ is

independent with ∘ ( ) ,

<
(ii) the mixing weights , = 1, … , is independent with and the

thinning at time , and
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