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It is proposed a new estimation method for the parameter of an INAR(p) process based on third-order cumulants.Note that this estimation method does not assume any particular discrete distribution of the countingseries and of the innovation process. The results of a Monte Carlo study to investigate and compare theperformance of the estimator are presented and the method is applied to a set of real data.
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I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Parameter estimation using third-ordercumulants for INAR(p) processes
Isabel Silva1 Maria Eduarda Silva2,3
1Departamento de Engenharia Civil and CEC, Faculdade de Engenharia da Universidade do Porto
2Faculdade de Economia da Universidade do Porto
3Unidade de Investigao Matemtica e Aplicaes (UIMA), Universidade de Aveiro
Workshop on Integer-valued Time Series Modelling (WINTS09)
WINTS09 1 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Contents
Introduction High-order statistics INteger-valued AutoRegressive (INAR) process
Parameter estimation based on third-order cumulants Cumulant third-order characterization of INAR(p) processes Estimation using cumulant Third-Order Recursion equation
Monte Carlo results and application to real data
Final remarks
WINTS09 2 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
High-Order Statistics (HOS)Moments and cumulants of order higher than two
Introduction WINTS09 3 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
High-Order Statistics (HOS)Moments and cumulants of order higher than two
Lack of Gaussianity and/or non-linearity
Introduction WINTS09 3 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
High-Order Statistics (HOS)Moments and cumulants of order higher than two
Lack of Gaussianity and/or non-linearity
Notation:{Xt} : kth-order stationary stochastic process
X(s1, . . . ,sk1) : kth-order joint moment of Xt,Xt+s1 , . . . , Xt+sk1function of k1 variables (s1, . . . ,sk1 R)
X = E[Xt], X(s1, . . . ,sk1) = E[XtXt+s1 . . .Xt+sk1 ]
CX(s1, . . . ,sk1) : kth-order joint cumulant of Xt,Xt+s1 , . . . , Xt+sk1function of k1 variables (s1, . . . ,sk1 R)
the coefficient of 12 . . .k in the Taylor series expansion (about (0, . . . ,0)) ofthe cumulant generating function of Xt,Xt+s1 , . . . , Xt+sk1
Introduction WINTS09 3 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
High-Order Statistics (HOS)Relations between joint moments and joint cumulants [Leonov and Shiryaev (1959)]CX = E[Xt] = XCX(k) = X(k)X2 = R(k), k ZCX(k,m) = X(k,m)X
(X(k)+ X(m)+ X(km)
)+2X3, k,m Z
Introduction WINTS09 4 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
High-Order Statistics (HOS)Relations between joint moments and joint cumulants [Leonov and Shiryaev (1959)]CX = E[Xt] = XCX(k) = X(k)X2 = R(k), k ZCX(k,m) = X(k,m)X
(X(k)+ X(m)+ X(km)
)+2X3, k,m Z
Symmetry properties [Mendel (1991)]
CX(m) = CX(m), m > 0CX(m,n) = CX(n,m) = CX(n,mn) = CX(nm,m), m,n > 0
n
m
0
Introduction WINTS09 4 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
INteger-valued AutoRegressive processes
INAR(p) [Latour (1998)]
Xt = 1 Xt1 +2 Xt2 + +p Xtp + et
Introduction WINTS09 5 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
INteger-valued AutoRegressive processes
INAR(p) [Latour (1998)]
Xt = 1 Xt1 +2 Xt2 + +p Xtp + et
0 i < 1, i = 1, . . . ,p1, and 0 < p < 1, such that pk=1 k < 1 thinning operation [Steutel and Van Harn (1979); Gauthier and Latour (1994)]
i Xti = Xtij=1 Yi,j, for i = 1, . . . ,p{Yi,j} (counting series): set of i.i.d. non-negative integer-valued r.v. withE[Yi,j] = i,Var[Yi,j] = 2i and E[Y3i,j] = i{et} (innovation process): sequence of i.i.d. non-negative integer-valued r.v.(independent of {Yi,j}) with E[et] = e,Var[et] = 2e and E[e3t ] = e
Introduction WINTS09 5 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
INteger-valued AutoRegressive processes
INAR(p) [Latour (1998)]
Xt = 1 Xt1 +2 Xt2 + +p Xtp + et
0 i < 1, i = 1, . . . ,p1, and 0 < p < 1, such that pk=1 k < 1 thinning operation [Steutel and Van Harn (1979); Gauthier and Latour (1994)]
i Xti = Xtij=1 Yi,j, for i = 1, . . . ,p{Yi,j} (counting series): set of i.i.d. non-negative integer-valued r.v. withE[Yi,j] = i,Var[Yi,j] = 2i and E[Y3i,j] = i{et} (innovation process): sequence of i.i.d. non-negative integer-valued r.v.(independent of {Yi,j}) with E[et] = e,Var[et] = 2e and E[e3t ] = e
Usually: Poisson INAR(p) process with binomial thinning operation
Introduction WINTS09 5 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Cumulant third-order characterization of INAR(p) processes
[Silva and Oliveira (2004, 2005) and Silva (2005)]CX(0,0) = pi=1 pj=1 pk=1 ijk X(i j, i k)+3pi=1 pj=1 ji2X(i j)+ e
+3(eX)pi=1 pj=1 ijX(i j)+3X(eX)pi=1 i2 +23X6eX2 pi=1 i3e(e2 +2e )+ X pi=1 (i3ii23i )
CX(0,k) = pi=1 iCX(0,k i), k > 0
CX(k,k) = pi=1 pj=1 ijCX(k i,k j)+pi=1 i2CX(k i), k > 0
CX(k,m) = pi=1 iCX(k,m i), m > k > 0
Parameter estimation based on third-order cumulants WINTS09 6 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Cumulant third-order characterization of INAR(p) processes
[Silva and Oliveira (2004, 2005) and Silva (2005)]CX(0,0) = pi=1 pj=1 pk=1 ijk X(i j, i k)+3pi=1 pj=1 ji2X(i j)+ e
+3(eX)pi=1 pj=1 ijX(i j)+3X(eX)pi=1 i2 +23X6eX2 pi=1 i3e(e2 +2e )+ X pi=1 (i3ii23i )
CX(0,k) = pi=1 iCX(0,k i), k > 0
CX(k,k) = pi=1 pj=1 ijCX(k i,k j)+pi=1 i2CX(k i), k > 0
CX(k,m) = pi=1 iCX(k,m i), m > k > 0
INAR processes have a non-linear structure
1st and 2nd order cumulants are not sufficient to describe dependence structure
Parameter estimation based on third-order cumulants WINTS09 6 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Cumulant third-order characterization of INAR(p) processes
Cumulant Third-Order Recursion (TOR) equation [Silva (2005)]
CX(k,m)p
i=1
iCX(i k, im) = (k)p
i=1
i2CX(im), 0 k m,m 6= 0
where (a) ={
1 if a = 0,0 otherwise,
is the Kronecker delta function
Parameter estimation based on third-order cumulants WINTS09 7 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Cumulant third-order characterization of INAR(p) processes
Cumulant Third-Order Recursion (TOR) equation [Silva (2005)]
CX(k,m)p
i=1
iCX(i k, im) = (k)p
i=1
i2CX(im), 0 k m,m 6= 0
where (a) ={
1 if a = 0,0 otherwise,
is the Kronecker delta function
~w CX(k,k) = CX(0,k)
C3,X =
CX(0,0) CX(1,1) CX(p1,p1)CX(0,1) CX(0,0) CX(p2,p2)
.
.
.
.
.
.
.
.
.
.
.
.
CX(0,p1) CX(0,p2) CX(0,0)
1
2.
.
.
p
=
CX(0,1)CX(0,2)
.
.
.
CX(0,p)
= c3,X
Parameter estimation based on third-order cumulants WINTS09 7 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Estimation using cumulant TOR equation
{X1, . . . ,XM,XM+1, . . . ,X2M, . . . ,X(B1)(M+1), . . . ,XN=BM}
Parameter estimation based on third-order cumulants WINTS09 8 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Estimation using cumulant TOR equation
{X1, . . . ,XM 1st block
,XM+1, . . . ,X2M 2nd block
, . . . ,X(B1)(M+1), . . . ,XN=BM Bth block
}
B blocks of M observations each
{X(1)1 , . . . ,X(1)M
1st block
,X(2)1 , . . . ,X(2)M
2nd block
, . . . ,X(B)1 , . . . ,X(B)M
Bth block
}
Parameter estimation based on third-order cumulants WINTS09 8 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Estimation using cumulant TOR equation
{X1, . . . ,XM 1st block
,XM+1, . . . ,X2M 2nd block
, . . . ,X(B1)(M+1), . . . ,XN=BM Bth block
}
B blocks of M observations each
{X(1)1 , . . . ,X(1)M
1st block
,X(2)1 , . . . ,X(2)M
2nd block
, . . . ,X(B)1 , . . . ,X(B)M
Bth block
}
For each block: X(i) = 1M
M
j=1
X(i)j
C(i)X (k,k) =1M
Mkj=1
(X(i)j X
(i))(
X(i)j+kX(i))2
, k = 0, . . . ,p1
C(i)X (0,k) =1M
Mkj=1
(X(i)j X
(i))2(
X(i)j+kX(i)), k = 1, . . . ,p
Parameter estimation based on third-order cumulants WINTS09 8 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Estimation using cumulant TOR equation
Overall third-order cumulant estimators:
CX(k,k) =1B
B
i=1
C(i)X (k,k), k = 0, . . . ,p1
CX(0,k) =1B
B
i=1
C(i)X (0,k), k = 1, . . . ,p
Parameter estimation based on third-order cumulants WINTS09 9 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Estimation using cumulant TOR equation
Overall third-order cumulant estimators:
CX(k,k) =1B
B
i=1
C(i)X (k,k), k = 0, . . . ,p1
CX(0,k) =1B
B
i=1
C(i)X (0,k), k = 1, . . . ,p
To solve the system of linear equations in order to the coefficients
CX(0,0) CX(1,1) CX(p1,p1)CX(0,1) CX(0,0) CX(p2,p2)
.
.
.
.
.
.
.
.
.
.
.
.
CX(0,p1) CX(0,p2) CX(0,0)
1
2.
.
.
p
=
CX(0,1)CX(0,2)
.
.
.
CX(0,p)
Parameter estimation based on third-order cumulants WINTS09 9 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Estimation using cumulant TOR equation
Overall third-order cumulant estimators:
CX(k,k) =1B
B
i=1
C(i)X (k,k), k = 0, . . . ,p1
CX(0,k) =1B
B
i=1
C(i)X (0,k), k = 1, . . . ,p
To solve the system of linear equations in order to the coefficients
CX(0,0) CX(1,1) CX(p1,p1)CX(0,1) CX(0,0) CX(p2,p2)
.
.
.
.
.
.
.
.
.
.
.
.
CX(0,p1) CX(0,p2) CX(0,0)
1
2.
.
.
p
=
CX(0,1)CX(0,2)
.
.
.
CX(0,p)
e = X(
1p
i=1
i
), 2e = VpX
p
i=1
2i , Vp = R(0)p
i=1
i R(i)
Parameter estimation based on third-order cumulants WINTS09 9 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Monte Carlo results
To examine the small sample properties of the proposed estimation method
To compare its performance with other methods: YW, CLS, WHT and LS_HOS
Monte Carlo results and application to real data WINTS09 10 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Monte Carlo results
To examine the small sample properties of the proposed estimation method
To compare its performance with other methods: YW, CLS, WHT and LS_HOS
1000 realizations of Poisson INAR(p) processes, with binomial thinning operation,for several orders, sample sizes and parameter values
Monte Carlo results and application to real data WINTS09 10 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Monte Carlo results
To examine the small sample properties of the proposed estimation method
To compare its performance with other methods: YW, CLS, WHT and LS_HOS
1000 realizations of Poisson INAR(p) processes, with binomial thinning operation,for several orders, sample sizes and parameter values
Sample properties of the cumulant TOR estimatorThe sample bias, variance and mean square error decrease as the sample size (ofthe block) increases Distribution is consistent and symmetric
Monte Carlo results and application to real data WINTS09 10 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Monte Carlo results
To examine the small sample properties of the proposed estimation method
To compare its performance with other methods: YW, CLS, WHT and LS_HOS
1000 realizations of Poisson INAR(p) processes, with binomial thinning operation,for several orders, sample sizes and parameter values
Sample properties of the cumulant TOR estimatorThe sample bias, variance and mean square error decrease as the sample size (ofthe block) increases Distribution is consistent and symmetricFor small sample size: evidence of departure from symmetry in the marginaldistributions, specially for values of the parameter near the non-stationary region
Monte Carlo results and application to real data WINTS09 10 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Monte Carlo results
To examine the small sample properties of the proposed estimation method
To compare its performance with other methods: YW, CLS, WHT and LS_HOS
1000 realizations of Poisson INAR(p) processes, with binomial thinning operation,for several orders, sample sizes and parameter values
Sample properties of the cumulant TOR estimatorThe sample bias, variance and mean square error decrease as the sample size (ofthe block) increases Distribution is consistent and symmetricFor small sample size: evidence of departure from symmetry in the marginaldistributions, specially for values of the parameter near the non-stationary region
Coefficients underestimated; innovation parameter overestimated
Monte Carlo results and application to real data WINTS09 10 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Monte Carlo results
To examine the small sample properties of the proposed estimation method
To compare its performance with other methods: YW, CLS, WHT and LS_HOS
1000 realizations of Poisson INAR(p) processes, with binomial thinning operation,for several orders, sample sizes and parameter values
Sample properties of the cumulant TOR estimatorThe sample bias, variance and mean square error decrease as the sample size (ofthe block) increases Distribution is consistent and symmetricFor small sample size: evidence of departure from symmetry in the marginaldistributions, specially for values of the parameter near the non-stationary region
Coefficients underestimated; innovation parameter overestimated
Fixed M statistics measures decrease as B increases and vice-versa
Monte Carlo results and application to real data WINTS09 10 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Monte Carlo results
TOR_1B TOR_2B TOR_4B TOR_1B TOR_2B TOR_4B2
1
0
1
2Bi
as(
1)
TOR_1B TOR_2B TOR_4B TOR_1B TOR_2B TOR_4B2
1
0
1
2
Bias
(2)
TOR_1B TOR_2B TOR_4B TOR_1B TOR_2B TOR_4B7
0
7
Bias
()N=60 N=200
Figure: Boxplots of the sample bias for the estimates obtained in 1000 realizations of 60 and 500 observations of theINAR(2) model: Xt = 0.6Xt1 +0.1Xt2 + et, where et Po(1).
Monte Carlo results and application to real data WINTS09 11 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Monte Carlo results
WHT CLS LS_HOS YW TOR_1B WHT CLS LS_HOS YW TOR_1B0.8
0.4
0
0.4
0.8Bi
as(
)
WHT CLS LS_HOS YW TOR_1B WHT CLS LS_HOS YW TOR_1B5
2.5
0
2.5
5
Bias
()(, )=(0.4, 3.0) (, )=(0.9, 3.0)
Figure: Boxplots of the sample bias for the estimates obtained in 1000 realizations of 200 observations of the INAR(1)models: Xt = 0.4Xt1 + et and Xt = 0.9Xt1 + et, where et Po(1).
Monte Carlo results and application to real data WINTS09 12 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Application to real data
1920 1930 1940 1950 1960 1970 19815
10
15
20
25
30
35
40
45
50
Num
ber o
f pla
nts
Figure: The number of Swedish mechanical paper and pulp mills, from 1921 to 1981 [Brnns (1995) and Brnns andHellstrm (2001)]
Monte Carlo results and application to real data WINTS09 13 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Application to real dataSimple INAR(1)It is not assumed the Poisson distribution for the innovation process:
X = 20.40 and S2 = 155.16
Monte Carlo results and application to real data WINTS09 14 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Application to real dataSimple INAR(1)It is not assumed the Poisson distribution for the innovation process:
X = 20.40 and S2 = 155.16
Method e 2e x 2x MSECLS 0.9591 0.2017 15.2268 4.9279 192.2764 9.3254
LS_HOS 0.9269 1.3635 19.2253 18.6516 145.4513 9.2997TOR_1B 0.9631 0.7518 14.7219 20.374 213.5073 8.9224
Monte Carlo results and application to real data WINTS09 14 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Application to real dataSimple INAR(1)It is not assumed the Poisson distribution for the innovation process:
X = 20.40 and S2 = 155.16
Method e 2e x 2x MSECLS 0.9591 0.2017 15.2268 4.9279 192.2764 9.3254
LS_HOS 0.9269 1.3635 19.2253 18.6516 145.4513 9.2997TOR_1B 0.9631 0.7518 14.7219 20.374 213.5073 8.9224
Mean and variance of the estimated models: x =e
1 and 2x =
(1 )(e + 2e )(1 )2(1+ )
Monte Carlo results and application to real data WINTS09 14 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Application to real dataSimple INAR(1)It is not assumed the Poisson distribution for the innovation process:
X = 20.40 and S2 = 155.16
Method e 2e x 2x MSECLS 0.9591 0.2017 15.2268 4.9279 192.2764 9.3254
LS_HOS 0.9269 1.3635 19.2253 18.6516 145.4513 9.2997TOR_1B 0.9631 0.7518 14.7219 20.374 213.5073 8.9224
Mean and variance of the estimated models: x =e
1 and 2x =
(1 )(e + 2e )(1 )2(1+ )
MSE between the observations and the fitted models based on TOR_1B, LS_HOS andCLS estimates
Monte Carlo results and application to real data WINTS09 14 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Application to real data
1920 1930 1940 1950 1960 1970 19815
10
15
20
25
30
35
40
45
50
years
Num
ber o
f pla
nts
real dataCLSTOR_1B
Figure: The number of plants and the fitted values considering the TOR_1B and CLS estimatesMonte Carlo results and application to real data WINTS09 15 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Final remarks
Advantage of HOS: capability to detect and characterize the deviations fromGaussianity and non-linearity of the processes
Final remarks WINTS09 16 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Final remarks
Advantage of HOS: capability to detect and characterize the deviations fromGaussianity and non-linearity of the processes
INAR processes are non-Gaussian
Parameter estimation method: Estimation using cumulant TOR equation
Final remarks WINTS09 16 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Final remarks
Advantage of HOS: capability to detect and characterize the deviations fromGaussianity and non-linearity of the processes
INAR processes are non-Gaussian
Parameter estimation method: Estimation using cumulant TOR equation
The method does not assume any particular discrete distribution for the countingseries and for the innovation process
Final remarks WINTS09 16 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Final remarks
Advantage of HOS: capability to detect and characterize the deviations fromGaussianity and non-linearity of the processes
INAR processes are non-Gaussian
Parameter estimation method: Estimation using cumulant TOR equation
The method does not assume any particular discrete distribution for the countingseries and for the innovation process
Monte Carlo results: cumulant TOR estimates provides acceptable results, interms of sample bias, variance and mean square error
Final remarks WINTS09 16 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
Final remarks
Advantage of HOS: capability to detect and characterize the deviations fromGaussianity and non-linearity of the processes
INAR processes are non-Gaussian
Parameter estimation method: Estimation using cumulant TOR equation
The method does not assume any particular discrete distribution for the countingseries and for the innovation process
Monte Carlo results: cumulant TOR estimates provides acceptable results, interms of sample bias, variance and mean square error
When used in the context of a non-Poisson real dataset the estimates that useHOS information provide a model with mean, variance and autocorrelationscloser to the sample values
Final remarks WINTS09 16 / 17
I. Silva and M. E. Silva Parameter estimation using third-order cumulants for INAR(p) processes
References
BRNNS, K. (1995).Explanatory Variables in the AR(1) Count Data Model.Ume Economic Studies 381.
BRNNS, K. and HELLSTRM, J. (2001).Generalized Integer-Valued Autoregression.Econometric Reviews 20 (4), 425-443.
GAUTHIER, G. and LATOUR, A. (1994).Convergence forte des estimateurs des paramtres dunprocessus GENAR(p).Annales des Sciences Mathmatiques du Qubec 18,49-71.
LATOUR, A. (1998).Existence and stochastic structure of a non-negativeinteger-valued autoregressive process.Journal of Time Series Analysis 19, 439-455.LEONOV, V.P. and SHIRYAEV, A.N. (1959).On a method of calculation of semi-invariants.Theory of Probability and its Applications (translated byBrown, J.R.) IV, pp. 319-329.
MENDEL, J. M. (1991).Tutorial on higher-order statistics (spectra) in signalprocessing and system theory: theoretical results and someapplicationsProceedings of the IEEE 79, pp. 278-305.SILVA, I. (2005).Contributions to the analysis of discrete-valued timeseries.PhD Thesis. Universidade do Porto, Portugal.
SILVA, M. E. and OLIVEIRA, V. L. (2004).Difference equations for the higher-order moments andcumulants of the INAR(1) model.Journal of Time Series Analysis 25, 317-333.SILVA, M. E. and OLIVEIRA, V. L. (2005).Difference equations for the higher-order moments andcumulants of the INAR(p) model.Journal of Time Series Analysis 26, 17-36.STEUTEL, F. W. and VAN HARN, K. (1979).Discrete analogues of self-decomposability and stability.The Annals of Probability 7, 893-899.
References WINTS09 17 / 17
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