Page 120 - Contributed Paper Session (CPS) - Volume 2

CPS1458 KHOO W.C et al.

A new mixed INAR(p) model for time series of
counts
1
2
KHOO Wooi Chen , ONG Seng Huat
1 Department of Applied Statistics, School of Mathematical Sciences, Sunway University,
Malaysia
2 Institute of Mathematical Sciences, University Malaya, Malaysia
Abstract
This paper investigates a new mixed AR (p model for time series of count.
)
Some structural properties such as conditional moments and autocorrelation
have been derived. For model fitting maximum likelihood estimation via the
Expectation-Maximization (EM) algorithm is used to estimate the parameters.
A real life crime data set is used to demonstrate the practical performance of
the proposed model.

Keywords
Mixture model; Thinning operator; Discrete-valued time series; Maximum
likelihood estimation

1. Model Background
Box and Jenkins’ (1976) ARMA processes have been widely applied for real
data in continuous time series community. However, the theory of ARMA
models is no longer applicable to discrete time series because the discreteness
of the values is not preserved. Discrete-time series modelling has been
receiving much interest over the past thirty years. There are many real life
discrete data, such as the number of insurance claims, number of abstract
reviews, frequency of crime and so on. Recently a discrete-valued time series
model has been constructed by Khoo et al. (2017) as a mixture of Pegram and
thinning (MPT) models. The motivation for proposing the MPT model is that
it provides much flexibility for modelling time series of count data. The first
order MPT (MPT(1)) model is defined as follows. Consider two independent

non-negative integer-valued variables  X 1 − t and  , the initial value of the
t
process X 0 , has initial distribution (XP 0 = i ) = 0 , then for every
t   ,0  , 1  , 2 ... , the MPT(1) process is defined by

1−

X = (   X t 1 − ) ( ,  t ) (1)
,
t

Where the mixing weight   ( ) 1,0 , and the innovation term  is a
t
sequence of identical and independent distributed (i.i.d) non-negative
integer-valued random variables and mutually independent with mean 

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