1. science
2. mathematics
3. statistics

The combined Poisson INMA(q) models for time series of counts
Kaizhi Yu1, Hong Zou2
1 Statistics School, Southwestern University of Finance and Economics, Chengdu, 611130, China
2 School of Economics, Southwestern University of Finance and Economics, Chengdu, 611130, China
Corresponding author email: [email protected]
Abstract. A new stationary qth-order integer-valued moving average process with Poisson
innovation is introduced based on decision random vector. Some statistical properties of the
process are established. Estimators of the parameters of the process are obtained using the method
of moments. Some numerical results of the estimators are presented to assess the performance of
moment estimators.
Keywords. Poisson; INMA; Stationary; Ergodicity; Multinomial distribution.
1. Introduction
In natural and social sciences, time series of correlated counting are met very often. Especially
in economics and medicine many interesting variables are nonnegative count data. For example:
the number of shareholders in large Finnish and Swedish stocks, big numbers even for frequently
traded stocks, the number of arrivals per week to the emergency service of the Hospital, monthly
polio incidence counts in Germany. Most of the research on count processes assume that the count
data are independent and identically distributed. However, in practice, observations may be
autocorrelated, and this may adversely affect the performance of traditional model developed
under the assumption of independence. In recent years, count data time series models have been
devised to avoid making restrictive assmptions on the distribution of the error term. Regression
models for time series count data have been proposed (Zeger, 1988; Brannas and Johansson, 1994).
On the other hand, there have been attempts to develop suitable classes of models that resemble
the structure and properties of the usual linear ARMA models. For instances, Al-Osh and Alzaid
(1988) proposed an integer-valued moving average model (INMA) for discrete data. An integervalued GARCH model was given to study overdispersed counts (Ferland et al., 2006). Silva et
al.(2009) considered the problem of forecasting in INAR(1) model. Random coefficient INAR
models were introduced by Zheng et al. (2006,2007). The Signed thinning operator was developed
by Kachour and Truquet (2011). A new stationary first-order integer-valued autoregressive
process with geometric marginal distribution based on the generalized binomial thinning was
introduced by Ristić, Nastić and Ilić (2013). In these analysis of counts, the class of integer-valued
moving average models plays an important role.
The nonnegative integer-valued moving average process of the order q ( INMAR(q) ) was introduced
by Al-Osh and Alzaid (1988). The INMA(q) process is defined by the recursion,
X t  1 t  t 1     q t  t  q   t
(1)
1
Where
 k t  t k   i 1 vi , and the {vi } designated by counting series, is a sequence of i.i.d.
 t k
bernoulli random variables with E (vi )   , independent of
 t  k . The parameters 1 ,...,  q 
(0,1). The thinning operation ‘ t ’ indicates the corresponding thinning is associated with time.
The term
i t  t and  j t  t are independent. The choice of appropriate marginal distributions.
is still problematic for getting a particular distribution of X t . To overcome these difficulties,
Weiß (2008) introduced a combined INAR(p) models by using ‘decision’ random variables. Ristić
and Nastić (2012) considered an piecewise functions for count data model. Therefore, we adopt the
similar approach to deal with the problem in INMA(q) models. In this article, we propose an
combined INMA(q) model by allowing the parameters value are vary with ‘decision’ random
vector.
The article is organized as follows: In Sections 2, we specify the model and derive some statistical
properties. Section 3 concerns unknown parameter estimation by Yule-Walker method. In Section 4,
we conduct some Monte Carlo simulations. Finally, Section 5 concludes.
2. Definition and Basic Properties of the PCINMA(q) Process
Definition 2.1 (PCINMA(q) Model). A stochastic process { X t } is said to be the Poisson combined
INMA(q) process if it satisfies the following recursive equations:
X t  Dt ,1 (  t  t 1 )    Dt ,q (  t  t  q )   t
(2)
where { t ; t } is a sequence of independent and identically distributed Poisson random variables
with parameter
 and   (0,1) . {Dt ; t  } be an i.i.d process of ‘decision’ random vector
Dt  ( Dt ,1 ,, Dt ,q ) ~ Mult (1; 1 ,, q ) , independent of { t ; t } . Moreover, the counting
series
 t  t k , k=1, … , q are independent of {Dt } and  t at time t . To our knowledge, few
efforts has been devoted to studying combined INMA models. In this article, we aim to fill in this
gap. The Definition 2.1 shows that X t is equal to
 t  t 1   t with probability 1 , or equal to
 t  t 2   t with probability 2 , … , else equal to  t  t q   t with probability q
( 1  1  2    q 1 ) .
The moments will be useful in obtaining the appropriate estimating equations for parameter
estimation.
2
Theorem 2.1. The numerical characteristics of { X t } are as follows.
(i)
 X : E ( X t )   (  1) ,
(8)
(ii)
 X2 : Var ( X t )   (1   )  ( 2  1) 2 ,
(9)
q
 2
2 2





 j j  k ,1  k  q  1

k

j  k 1

 2
(iii)  X (k ) : cov( X t , X t  k )   k , k  q
0, k  q  1


(10)
Proof.
(i) E ( X t )  E[ Dt ,1 (  t  t 1   t )]    E[ Dt , q (  t  t  q   t )]
 E ( Dt ,1 ) E (  t  t 1   t )    E ( Dt ,q ) E (  t  t  q   t )
 1 (   )    q (   )   (  1)
(ii) Moreover,
 q

Var ( X t )  Var   Dt , j (  t  t  j   t ) 
 j 1

  Var  Dt , j (  t  t  j   t )   2 cov  Dt ,i (  t  t i   t ), Dt , j (  t  t  j   t ) 
q
j 1
i j

Note that Var Dt , j (  t  t  j   t )

in first summation on the right-hand side is
 Var ( Dt , j ) E (  t  t  j   t ) 2  E 2 ( Dt , j )Var (  t  t  j   t )
  j (1   j ) E  (  t  t  j ) 2   t2  2(  t  t  j ) t 
 j2 Var (  t  t  j )  Var ( t )  2 cov(  t  t  j ,  t ) 
  j (1   j ){[ 2 E ( t2 j )   (1   ) E ( t  j )]  E ( t2 )  2 E (  t  t  j ) E ( t )}
 j2 {[ 2Var ( t  j )   (1   ) E ( t  j )]   2 }
  j (1   j )[(  1) 2  2   (1   )  ( 2  1) 2 ]   j2 [( 2  1) 2   (1   ) ]
  j (1   j )(  1) 2  2   j [ (1   )  ( 2  1) 2 ]
3
By the same arguments as above, it follws that
cov  Dt ,i (  t  t i   t ), Dt , j (  t  t  j   t ) 
 cov  Dt ,i (  t  t i ), Dt , j (  t  t  j )   cov  Dt ,i (  t  t i ), Dt , j  t 
 cov  Dt ,i t , Dt , j (  t  t  j )   cov  Dt ,i t , Dt , j  t 
 cov( Dt ,i , Dt , j ) E (  t  t i ) E (  t  t  j )  cov( Dt ,i , Dt , j ) E (  t  t i ) E ( t )
 cov( Dt ,i , Dt , j ) E ( t ) E (  t  t  j )  cov( Dt ,i , Dt , j ) E 2 ( t )
 i j ( 2  2   2   2   2 )  i j (  1) 2  2
Therefore,
 q

Var ( X t )  Var   Dt , j (  t  t  j   t ) 
 j 1

   j (1   j )(  1)2  2   j [ (1   )  ( 2  1) 2 ]  2 i j (  1)2  2
q
j 1
i j
q


 (  1) 2  2 1    j2   [ (1   )  ( 2  1) 2 ]  2(  1) 2  2  i j
j 1
i j


  (1   )  ( 2  1) 2
(iii) For 1  k  q  1 , the autocovariance function of X t is
q
 q

cov( X t , X t  k )  cov   Dt ,i (  t  t i   t ),  Dt  k , j (  t  k  t  k  j   t  k ) 
j 1
 i 1

q


 cov  Dt ,k (  t  t  k   t ),  Dt  k , j (  t  k  t  k  j   t  k ) 
j 1


  cov  Dt , j (  t  j  t  j   t ), Dt  k , j  k (  t  k  t  k ( j  k )   t  k ) 
q
j  k 1
  cov  Dt ,k (  t  t  k ), Dt  k , j  t  k  
q
j 1
 cov  D
q
j  k 1
t, j
(  t  j  t  j ), Dt  k , j  k (  t  k  t  j ) 
q
  cov( t  t  k ,  t  k ) E ( Dt ,k ) E ( Dt  k , j )
j 1
q
  cov( t  j  t  j ,  t  k  t  j ) E ( Dt , j ) E ( Dt  k , j  k )
j  k 1
4
q
  Var ( t  k )k j 
j 1
  k   
2
2
2
q
  Var (
2
j  k 1
t j
) j j  k
q

j  k 1
j
j k
2
If k  q , by using a similar approach,we get cov( X t , X t  q )   q . For k  q  1 , all the
terms,
 t  t 1 , … ,  t  t q ，  t ，  t k  t k 1 , …,  t k  t k q and  t k involved in X t
and X t  k , are mutually independent. Therefore, the autocovariance function of X t is equal to
zero for k  q  1 .
Theorem 2.2. Let X t be the process defined by the equation in (2). Then
(i)
The { X t } is a covariance stationary process.
(ii) E ( X t )  C   , k  1, 2, 3,4 for some constant C  0 .
k
Proof. (i) The first conclusion is immediate from the definition of covariance stationary process.
(ii) For k  1, it is straightforward.
For k  2, it follows that

E ( X t2 )  max E  Dt ,1 (  t  t 1   t )  , , E  Dt ,q (  t  t  q   t ) 
2
2

Note that, for j  1, 2,..., q
E  Dt , j (  t  t  j   t )   E ( Dt , j )2 E (  t  t  j   t )2
2
 E ( Dt , j ) 2  E (  t  t  j ) 2  E ( t ) 2  2 E[(  t  t  j ) t ]
  j (1   j )[ 2 (   2 )   (1   )  (   2 )  2 2 ]
  j (1   j )(  1)[(  1)  1]
Then we have ,
1
(  1)[(  1)  1]  
4
Similarly, for k  3
E ( X t2 ) 

E ( X t3 )  max E  Dt ,1 (  t  t 1   t )  ,, E  Dt ,q (  t  t  q   t ) 
3
3

Note that, for j  1, 2,..., q
5
E  Dt , j (  t  t  j   t )   E ( Dt , j )3 E (  t  t  j   t )3
3
 E ( Dt , j )3  E (  t  t  j )3  E ( t )3  3E[(  t  t  j ) 2  t ]  E[(  t  t  j ) t2 ]
  3j [ 3  3 2 (1   )(   2 )  (  3 2 (1   )   3 ) ]
 3j   3 3j {[ 2 (   2 )   (1   ) ]}  3 3j  (   2 )
  3j {[ 2 (  1  3 )  3 (1   )  3 2  6  1]   }
Where
3
 :  3  3 2   . Let max
 max(13 ,  , q3 ) , Then
3
E ( X t3 )  max
{[ 2 (  1  3 )  3 (1   )  3 2  6  1]   }   .
After some tedious calculaions, we also can show the result holds for k  4 . We skip the details.
Next, we will present ergodic theorem for stationary process { X t } . There are a variety of ergodic
theorems, differing in their assumptions and in the modes of convergence. Here the convergence is
in mean square. The next two lemmas will be useful in proving the ergodicity of the sample mean
and smple autocovariance function of { X t } .
Lemma 2.1. If {Zt } is stationary with mean
Z and autocovariance function  Z () , then as
T   , Var ( ZT )  E ( ZT   Z ) 2  0 , if  Z (T )  0 .where ZT :
1 T
 Xt .
T t 1
Proof. See Theorem 7.1.1 in Brockwell, Peter J., and Richard A. Davis (2009) .
Lemma 2.2. Suppose {Zt } is a covariance stationary process with zero mean and covariance
function
1 T
[ E ( Z t Z t  v Z t l Z t l  v )   Z2 (v)]  0 .

T  T
l 1
 Z (v) : E ( Zt v Zt ) and. If lim
Then, for any fixed, v  0, 1, 2, …,
lim
T 
1 T
E[ Rˆ Z (v )  RZ2 (v )]2  0

l 1
T
Where ˆZ (v) is the sample covariance function ˆZ (v ) :
1 T v
 Zt v Zt .
T t 1
Proof. See Theorem 5.2 in Karlin, Samuel, and Howard E. Taylor (1975) .
Lemma 2.3. Let process Yt : X t   X is a transformation of X t , then the following results
hold.
(i)
{Yt } is a covariance stationary process with zero mean;
6
(ii)
 Y (k ) : cov(Yt , Yt k )   X (k ) , k  1, 2, 3,…;
(iii) E (Yt )  C   , k  1, 2, 3,4 for some constant C  0
k
*
*
(iv) {Yt } is ergodic in autocovariance function.
Proof. The only part of this lemma that is not obvious is part (iv). The proof of properties (iv) is
follows.
To prove that {Yt } is ergodic in autocovariance function, it suffices to show that
1 T
[ E (YtYt  vYt lYt l  v )   Y2 (v)]  0 according to lemma 2.2. Thus we will discuss two

T  T
l 1
lim
cases. For simplicity in notation we define c2 : E (Yt ) , c3 : E (Yt ) , c4 : E (Yt ) ,
2
3
4
and RY (v) : E (YY
t t v ) .
Case 1. For v  0 , E (YtYt  vYt lYt l  v )  E (Yt Yt l ) ，  Y (v)   X (v)   X
2
2
2
2
2
If 1  l  q , using the schwarz inequality we get
E (Yt 2Yt 2 l )  E (| Yt 2 |2 ) E (| Yt 2 l |2 )  c4  C *
If l  q  1 , note that Yt l and Yt are irrelevant, then
E (Yt 2Yt 2 l )  E (Yt 2 ) E (Yt 2 l )  c22   X2 (v)
Therefore,
1 T
1
[ E (YtYt  vYt lYt l  v )   Y2 (v)]  [2(c4   X2 )]  0 ，for T  

T l 1
T
Case 2. For v  1 , RY (v)  [ E (YtYt  v )]   X (v)
2
2
2
If 1  l  q  v , using schwarz inequality twice,
E (YY
E (| YtYt  v |2 ) E (| Yt lYt l  v |2 )
t t  vYt l Yt l  v ) 
1
 4 E (Yt 4 ) E (Yt 4 v ) E (Yt 4 l ) E (Yt 4 l  v )  c4
If l  q  v  1 时，note that X t X t v 与 X t l X t l  v uncorrelation，thus we have
2
E (YY
t t  vYt l Yt l  v )  E ( X t X t  v ) E (Yt  l Yt  l  v )  RY (v )
Then
1 T
1
[ E (YtYt  vYt lYt l  v )   Y2 (v)]  {(2  v)[c4  RY2 (v)]}  0 , for T   .

T l 1
T
7
This proves Lemma 2.3.
Theorem 2.3. Let { X t } be a PCINMA(q) process according to Definition 2.1. Then, the
stochastic process { X t } is ergodic in the mean and autocovariance function.
Proof. For notational simplicity, we define X T (k ) :
ˆX (k ) :
1 T k
1 T k
ˆ
X

k

Yt  k Yt ,
 t k , Y ( ) : T 
T t 1
t 1
1 T k
 ( X t k  X T )( X t  X T ) . And we assume here that the sample consists of T  k
T t 1
observations on X .
(i) Note that
 X (k )  0 , for k   . From the result of lemma 2.1, we obtain
Var ( X T )  E ( X T   X ) 2  0 .
Then X T converges in probability to
 X . Therefore, the process { X t } is ergodic in the mean.
Next, we prove the { X t } is ergodic for second moment by induction.
  X . Suppose
(ii) First we prove X T (k ) 
P
P (| X T ( k )   X |  1 )  P (| X T ( k )  X |
 P (|
1
2
1  0 is given.
)  P (| X T   X |
1
2
)


1 k
X t | 1 )  P (| X T   X | 1 )

2
2
T t 1
k


1
  P ( | X t | 1 )  P (| X T   X | 1 )
2
2
T
t 1
k
E( X t )
1
1


 0 , for T   .
P
(
|
X
|
)


t
1
T
2
t 1
t 1 2 T  1
k
Using the Markov inequality,
Scince { X t } is ergodic in the mean, thus P (| X T   X |
1
2
)  0 , for T   .
  X , for T   .
Therefore, X T (k ) 
P
P
Now we prove the second result ˆ X ( k )   X ( k ) 
0 .
Consider any
  0.
P(| ˆX (k )   X (k ) |  )  P(| ˆX (k )  ˆY (k )  ˆY (k )   X (k ) |  )


 P(| ˆX (k )  ˆY (k ) | )  P(| ˆY (k )   X (k ) | )
2
2
8
Note that


P(| ˆX (k )  ˆY (k ) | )  P(| ( X T (k )  X T )( X T   X )  ( X T2   X2 ) | )
2
2


 P (| ( X T ( k )  X T )( X T   X ) | )  P (| X T2   X2 | )
4
4
Since the sample mean X T converges in probability to
 X . According to the slutsky’s theorem,
 0 , X T   X 
0 .
we get ( X T (k )  X T )( X T   X ) 
P
Then we have P (| ˆX ( k )  ˆY (k ) |

2
2
2
P
P
) 
 0 , for T   .
From the (iv) of lemma 2.3, we obtain
P
ˆY (k )   X (k )  ˆY (k )   Y (k ) 
 0 , for T   .
 0 , for T  
And consequently, P(| ˆ X (k )   X (k ) |  ) 
P
This leads to the desired conclusion.
3. Estimation of the unknown parameters
In this section we discuss approaches to the estimation of the unknown parameters. And we assume
we have T observations, X 1 ,…, X T , from a Poisson combined INMA(q) process in which, the
order parameter, q is known. One of the main interests in the literature of INMA process is to
estimate the unknown parameters. Using the sample covariance function, we get the estimators of
unknown parameters ( 1 ,…,
q ,  ,  ) through solving the following equations.
ˆ (0)  [ (1   )  ( 2  1) 2 ]  0

q
ˆ (1)  ( 2   2 2   )  0

1
j j 1

j 2



ˆ (q  1)  ( 2   2 2  )  0
q 1
q 1

ˆ (q)   2  0
q


The idea behind these estimators is that of equating population moments to sample moments and
then solving for the parameters in terms of the sample moments. These estimators are typically
called the Yule-Walker estimators. As the ˆ (k ) consistently estimates the true autocovariance
function
 (k ) (Brockwell and Davis, 1991), the Yule-Walker estimators are consistent. Following
Brockwell and Davis (2009) and Billingsley (1961), it is easy to show that under some mild
9
moment conditions the marginal mean estimator X T is asymptotically normally distributed.
4. Monte Carlo simulation study
We provide some simulations results to show the empirical performance of these estimators.
Owing to the nonlinearity, the estimator expressions of unknown parameters are quite complicated.
The aim of simulation study is to assess the finite sample performances of the moments stimators.
Consider the following model
X t  Dt ,1 ( t  t 1 )  Dt ,2 ( t  t  2 )  Dt ,3 ( t  t 3 )   t
where { t } is a sequence of i.i.d Poisson random variables with parameter
 and   (0,1) .
The random vectors ( Dt ,1 , Dt ,2 , Dt ,3 ) is multinomial distribution with parameters (1; 1 , 2 , 3 ) ,
independent of { t } . The parameters values considered are:
(model A) (1 , 2 , 3 )  (0.1, 0.2, 0.7) ,   0.3 ,   1 ,
(model B) (1 , 2 , 3 )  (0.3,0.4,0.3) ,   0.5 ,   3 ,
(model C) (1 , 2 , 3 )  (0.6,0.2,0.2) ,   0.7 ,   10 .
The length of this discrete-valued time series T is 50, 100, 400 and 700.For each realization of
these estimators, 500 independent replicates were simulated. The numerical results of the estimators
for different true values of the parameters
(1 , 2 , 3 ) ,  , and  are presented in Table 1.
Table 1 Sample mean and mean square error (in brackets) for models A, B and C.
Sample size
Model
Parameter
1
2
A


1
B
2
50
100
400
700
0.0781
(0.0316)
0.1658
(0.0251)
0.2433
(0.0236)
1.1372
(0.1256)
0.0873
(0.0084)
0.1795
(0.0193)
0.2731
(0.0074)
1.1007
(0.0632)
0.0925
(0.0047)
0.1867
(0.0052)
0.2875
(0.0032)
1.0285
(0.0358)
0.0953
(0.0017)
0.1957
(0.0035)
0.2924
(0.0021)
1.0046
(0.0067)
0.2547
(0.0412)
0.3468
(0.0308)
0.2871
(0.0136)
0.3726
(0.0158)
0.2935
(0.0087)
0.3891
(0.0072)
0.2977
(0.0032)
0.3964
(0.0023)
10


1
2
C


All the biases of
0.4625
(0.0217)
3.5672
(0.3265)
0.4789
(0.0104)
3.1741
(0.2381)
0.4836
(0.0044)
3.0522
(0.0943)
0.4976
(0.0013)
3.0075
(0.0045)
0.5648
(0.0213)
0.1732
(0.0342)
0.6347
(0.0145)
12.3627
(1.0764)
0.5783
(0.0151)
0.1847
(0.0153)
0.6824
(0.0083)
10.6538
(0.3105)
0.5867
(0.0078)
0.1907
(0.0024)
0.6947
(0.0026)
10.1486
(0.0852)
0.5973
(0.0024)
0.1976
(0.0005)
0.6953
(0.0011)
10.0024
(0.0053)
1 , 2 and  are negative, whereas the biases of  is positive. It can be
seen that as the sample size increases, the estimates seem to converge to the true parameter values.
For example, when increasing sample size T, the Bias and MSE are both converges to zero. The
reason might be that the Yule-Walker method is based on sufficient statistics. On the other hand ,
The performance of the estimators of
 are weaker than for the ones of 1 and  .
5. Conclusion
In this article, we introduce a class of self-exciting threshold integer-valued Moving average
models driven by decision random vector. Basic probabilistic and statistical properties of this class
of models are discussed. Specifically, the method of estimation under analysis is the Yule-Walker.
Their performance is compared through a simulation study. Potential issues of future research include
to extend the results for general INARMA(p, q) models including an arbitrary distribution of
binomial thinning parameter as well as autoregressive and moving average parameters. This
remains a topic of future research.
6. Acknowledgements
We are grateful to the Professor Peter X.-K. Song from University of Michigan for his helps and
suggestions. We also thank an anonymous referee for helpful suggestions. This work is supported
by the National Natural Science Foundation of China (no. 71171166 and no. 71201126), the Science
Foundation of Ministry of Education of China (no. 12XJC910001), and the Fundamental Research
Funds for the Central Universities (no. JBK150949, JBK150501, JBK120405).
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