Presentation Seminar PDMA

8/6/2019 Presentation Seminar PDMA

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Isabel Silva Magalhes Integer-valued time series

Integer-valued time series

Isabel Silva Magalhes

Departamento de Engenharia Civil, Faculdade de Engenharia da Universidade do Porto

Unidade de Investigao Matemtica e Aplicaes (UIMA), Universidade de Aveiro

PDMA-UP - October 2009

PDMA-UP - October 2009 1 / 30


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Outline

Motivation

Thinning operation

INteger-valued AutoRegressive (INAR) processes

INAR(1) processes INAR(p) processes

Parameter estimation for INAR(p) processes

Application to real data

Recent developments

Theme proposal

PDMA-UP - October 2009 2 / 30


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Motivation

Discrete time non-negative integer-valued time series counting series

Motivation PDMA-UP - October 2009 3 / 30


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Motivation


Examples:

the daily number of seizures of epileptic patients

the yearly number of plants in a region

the daily number of guest nights in a hotel

the monthly incidence of a disease.


I b l Sil M lh I l d i i


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Motivation


Examples:





Usual linear time series models (ARMA processes) are not suitable.


Isabel Sil a Magalhes Integer al ed time series


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Motivation


Examples:





Usual linear time series models (ARMA processes) are not suitable.

Thinning operation

Multiplication counterpart on the integer-valued context.


Isabel Silva Magalhes Integer valued time series


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Thinning operation

Binomial thinning operation [Steutel and Van Harn (1979)]

X : non-negative integer-valued random variable (r.v.), 0

X=X

j=1

Yj

{Yj} N0 (counting series): sequence of independent and identically distributed(i.i.d.) r.v., independent ofX, P(Yj = 1) = 1P(Yj = 0) =

X is the number of successes, with probability , in X trials

Thinning operation PDMA-UP - October 2009 4 / 30



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Thinning operation

Binomial thinning operation [Steutel and Van Harn (1979)]

X : non-negative integer-valued random variable (r.v.), 0

X=X

j=1

Yj

{Yj} N0 (counting series): sequence of independent and identically distributed(i.i.d.) r.v., independent ofX, P(Yj = 1) = 1P(Yj = 0) =

X is the number of successes, with probability , in X trials

Generalized thinning operation [Gauthier and Latour (1994); Silva and Oliveira (2004, 2005)]

X=X

j=1

Yj

{Yj}: sequence of i.i.d.r.v., independent ofX, with some discrete distribution such thatE[Yj] = , Var[Yj] =

2, E[Yj3] = , E[Yj

4] =

Thinning operation PDMA-UP - October 2009 4 / 30



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INAR(1) processes [McKenzie (1985, 1988); Al-Osh and Alzaid (1987)]

Xt = Xt1 + et, t= 1, . . . ,N

INteger-valued AutoRegressive (INAR) processes PDMA-UP - October 2009 5 / 30



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g g


INAR(1) processes [McKenzie (1985, 1988); Al-Osh and Alzaid (1987)]

Xt = Xt1 + et, t= 1, . . . ,N

0 <

1

{et} N0 : sequence of i.i.d. discrete r.v. (arrival process)

E[et] = e, Var[et] = e2, E[et

3] = e, E[et4] = e

counting series, {Yj}, are independent, and independent of{et}, and such thatE[Yj] = , Var[Yj] =

2, E[Yj3] = , E[Yj

4] =




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Poisson INAR(1) process with binomial thinning operation

Xt = Xt1 + et

{et}

Poisson distributed and

{Yj}

Bernoulli distributed




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Xt = Xt1 + et

{et}


{Yj}


Xt1 : survivors of the elements of the process at time t1, each withprobability of survival

et : elements which enter in the system in the interval ]t

1, t]




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Xt = Xt1 + et

{et}


{Yj}


Xt1 : survivors of the elements of the process at time t1, each withprobability of survival

et : elements which enter in the system in the interval ]t

1, t]

X1 Po

1

{et} Po()

={Xt} Po

1




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Likelihood function of the Poisson INAR(1) process

p(Xt|Xt1) : convolution of binomial distribution and Poisson distributionp(Xt|Xt1) = exp()

Mt

i=0

(Xt)i

((Xt) i)!

Xt1i

i(1)(Xt1)i,Mt = min{Xt1,Xt}




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Mt

i=0

(Xt)i

((Xt) i)!

Xt1i


X = {X0,X1, . . . ,XN}L(X,,) =

[/(1)]X0X0!

exp

1

N

t=1

p(Xt|Xt1)

L(X,,|X0)




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Mt

i=0

(Xt)i

((Xt) i)!

Xt1i


X = {X0,X1, . . . ,XN}L(X,,) =

[/(1)]X0X0!

exp

1

N

t=1

p(Xt|Xt1)

L(X,,|X0)Poisson distribution mean = variance problem in practical applications




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Mt

i=0

(Xt)i

((Xt) i)!

Xt1i


X = {X0,X1, . . . ,XN}L(X,,) =

[/(1)]X0X0!

exp

1

N

t=1

p(Xt|Xt1)

L(X,,|X0)Poisson distribution mean = variance problem in practical applicationsOverdispersion: variance > mean binomial, negative binomial, geometric or

generalized Poisson

Underdispersion: variance < mean generalized PoissonINteger-valued AutoRegressive (INAR) processes PDMA-UP - October 2009 7 / 30



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INAR(p) processes [Du and Li (1991); Latour (1998)]

Xt = 1 Xt1 + +p Xtp + et, t= 1, . . . ,N,

i 0, i = 1, . . . ,p1, and p > 0, such that pi=1i < 1,{et} N0 : sequence of i.i.d. discrete r.v. (arrival process)


3] = e, E[et4] = e,

all counting series, {Yj,i}, of the thinning operationsi Xti =Xtij=0 Yj,i, i = 1, . . . ,p,

are mutually independent, and independent of{et}, and such thatE[Yj,i] = i, Var[Yj,i] = i

2, E[Yj,i3] = i, E[Yj,i

4] = i.




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INAR(p) processes [Du and Li (1991); Latour (1998)]

Xt = 1 Xt1 + +p Xtp + et, t= 1, . . . ,N,

i 0, i = 1, . . . ,p1, and p > 0, such that pi=1i < 1,{et} N0 : sequence of i.i.d. discrete r.v. (arrival process)


3] = e, E[et4] = e,

all counting series, {Yj,i}, of the thinning operationsi Xti =Xtij=0 Yj,i, i = 1, . . . ,p,

are mutually independent, and independent of{et}, and such thatE[Yj,i] = i, Var[Yj,i] = i

2, E[Yj,i3] = i, E[Yj,i

4] = i.

INAR(p) has the same second-order structure as an AR(p) process




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Autocovariance function [Gauthier and Latour (1994)]

R(0) = Vp +pi=1iR(i),R(k) =

pi=1iR(i k), k Z\{0},

Spectral density function [Silva and Oliveira (2004, 2005)]

f() = 12

Vp1pk=1keik2 , ,One-step-ahead prediction error [Silva (2005)]

Vp = e2 +Xpi=1i2.




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Autocovariance function [Gauthier and Latour (1994)]

R(0) = Vp +pi=1iR(i),R(k) =

pi=1iR(i k), k Z\{0},

Spectral density function [Silva and Oliveira (2004, 2005)]

f() = 12

Vp1pk=1keik2 , ,One-step-ahead prediction error [Silva (2005)]

Vp = e2 +Xpi=1i2.

One-sided general linear representation [Silva (2005)]

Xt =

u=0ut

u,

{t

}white noise process




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Third-order characterization (in terms of moments)[Silva and Oliveira (2004, 2005) and Silva (2005)]

X(0, 0) = pi=1

pj=1

pk=1ijkX(ij, i k) + 3pi=1pj=1ji2X(ij)

+3X(2e +e

2)pi=1i + 3e

pi=1

pj=1ijX(ij)

+3Xep

i=1i

2 +Xp

i=1(i

3ii2

3

i

) + e

X(0, k) = pi=1iX(0, k i) +eX(0), k> 0

X(k, k) = pi=1

pj=1ij X(k

i, k

j) +

pi=1i

2X(k

i) + 2eX(k)

X(e2e2), k> 0

X(k, m) = pi=1iX(k, m i) +eX(k), m > k> 0




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Third-order characterization (in terms of cumulants)[Silva and Oliveira (2004, 2005) and Silva (2005)]

CX(0, 0) = pi=1

pj=1

pk=1ijkX(ij, i k) + 3pi=1pj=1ji2X(ij) + e

+3(eX)pi=1pj=1ijX(ij) + 3X(eX)pi=1i2 + 23X6eX2pi=1i3e(e2 +2e ) +Xpi=1 (i3ii23i )

CX(0, k) = pi=1iCX(0, k i), k> 0

CX(k, k) = pi=1

pj=1ijCX(k i, kj) +pi=1i2CX(k i), k> 0

CX(k, m) = pi=1iCX(k, m i), m > k> 0




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Third-order characterization (in terms of cumulants)[Silva and Oliveira (2004, 2005) and Silva (2005)]

CX(0, 0) = pi=1

pj=1

pk=1ijkX(ij, i k) + 3pi=1pj=1ji2X(ij) + e

+3(eX)pi=1pj=1ijX(ij) + 3X(eX)pi=1i2 + 23X6eX2pi=1i3e(e2 +2e ) +Xpi=1 (i3ii23i )

CX(0, k) = pi=1iCX(0, k i), k> 0

CX(k, k) = pi=1

pj=1ijCX(k i, kj) +pi=1i2CX(k i), k> 0

CX(k, m) = pi=1iCX(k, m i), m > k> 0

INAR processes have a non-linear structure

1st and 2nd order moments and cumulants are not sufficient to describe dependence structure




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Some extensions of INAR processes and relations with other models

Switching INAR(1) process [Franke and Seligmann (1993)]

Inclusion of explanatory variables [Brnns (1995)]

Panel data [Brnns (1994, 1995)]

INteger-valued Moving Average (INMA) model [Al-Osh and Alzaid (1988); Mckenzie (1988);

Brnns and Hall (2001)]

INteger-valued AutoRegressive-Moving Average (INARMA) model [Mckenzie (1985, 1986);

Al-Osh and Alzaid (1991)]

Multivariate INAR(p) process [Franke and Subba Rao (1995); Latour (1997)]

INAR(p) of Alzaid and Al-Osh (1990) ARMA(p,p1)Poisson INAR(1) process is a M/M/ queueing system observed at regularly spaced interval of

times [Steutel et al. (1983); Mckenzie (1988)]

INAR(p) is a Multitype Branching processes with immigration, BGWI(p) [Dion et al. (1995)]




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Time domain

Method of Moments Second-order: Yule-Walker estimation (YW)

Third-order: Estimation using the Cumulant Third-Order Recursion equation

(TOR),

Least Squares (LS) estimation Second-order: Conditional Least Squares (CLS)

Third-order: LS estimation based on high-order statistics (LS_HOS),

Frequency domain

Whittle estimation (WHT),

Parameter estimation for INAR(p) processes PDMA-UP - October 2009 13 / 30



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Time domain

Method of Moments Second-order: Yule-Walker estimation (YW)

Third-order: Estimation using the Cumulant Third-Order Recursion equation

(TOR),

Least Squares (LS) estimation Second-order: Conditional Least Squares (CLS)

Third-order: LS estimation based on high-order statistics (LS_HOS),

Frequency domain

Whittle estimation (WHT),

YW, CLS, WHT, TOR, LS_HOS: do not assume the Poisson distribution for the

arrival process

more adaptive and flexible methods




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Theorem (Asymptotic distribution of the Yule-Walker estimators of an INAR(p) process [Silva and Silva (2006)])

Let {Xt} be an INAR(p) process and the Yule-Walker estimator of

[Al-Osh and Alzaid

(1987); Du and Li (1991)], that is

Rp

1= rp

R(0) R(1) R(p1)R(1) R(0) R(p2)

.

..

.

.... .

.

..R(p1) R(p2) R(0)

1

2.

..

p

=

R(1)

R(2).

..R(p)

.

Then

N1/2 () is AN(0p, Vyw),where 0n is a vector ofn zeros and Vyw = D

TRrD, for Rr given by

Rr(i,j) = Cov(VRr(i), VRr(j)) and DT =

R(1)Ip R(p)Ip

(Rp11)T

Rp11

Ip2 0p2p

+

0pp2 Rp1

1

, with In the nn identity matrix,

0nm

the n

m matrix of zeros and

the Kronecker.




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Theorem (Cumulant Third-Order Recursion (TOR) equation of INAR(p) processes [Silva (2005)])

Let {Xt} be a stationary INAR(p) process. Then the third-order cumulants ofXt canbe written in a single cumulant Third-Order Recursion (TOR) equation by

CX(k,m)pi=1iCX(i k, im) = (k)p

i=1i

2CX(im),

where 0 k m, m = 0, and (a) is the Kronecker delta function.



( )


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Theorem (Cumulant Third-Order Recursion (TOR) equation of INAR(p) processes [Silva (2005)])

Let {Xt} be a stationary INAR(p) process. Then the third-order cumulants ofXt canbe written in a single cumulant Third-Order Recursion (TOR) equation by

CX(k,m)pi=1iCX(i k, im) = (k)p

i=1i

2CX(im),

where 0 k m, m = 0, and (a) is the Kronecker delta function.

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



P i i f INAR( )


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Estimation using the Cumulant TOR equation

{X1, . . . ,XN=B M} = {X(1)1 , . . . ,X(1)M ,X(2)1 , . . . ,X(2)M , . . . ,X(B)1 , . . . ,X(B)M },

C(i)

X (k, k) =1

M

Mkj=1

(X(i)

j X(i))(X(i)j+kX(i)

)2, k= 0, . . . ,p1,

C

(i)

X (0, k) =

1

M

Mkj=1 (X

(i)

j X(i)

)

2

(X

(i)

j+kX(i)

), k= 1, . . . ,p,

CX(k, k) =1

B

B

i=1C

(i)X (k, k), k= 0, . . . ,p1,

CX(0, k) =1

B

B

i=1C

(i)X (0, k), k= 1, . . . ,p,

Solve C3,X= c3,X.



P t ti ti f INAR( )


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Estimation using the Cumulant TOR equation

{X1, . . . ,XN=B M} = {X(1)1 , . . . ,X(1)M ,X(2)1 , . . . ,X(2)M , . . . ,X(B)1 , . . . ,X(B)M },

C(i)

X (k, k) =1

M

Mkj=1

(X(i)

j X(i))(X(i)j+kX(i)

)2, k= 0, . . . ,p1,

C

(i)

X (0, k) =

1

M

Mkj=1 (X

(i)

j X(i)

)

2

(X

(i)

j+kX(i)

), k= 1, . . . ,p,

CX(k, k) =1

B

B

i=1C

(i)X (k, k), k= 0, . . . ,p1,

CX(0, k) =1

B

B

i=1C

(i)X (0, k), k= 1, . . . ,p,

Solve C3,X= c3,X.

Asymptotic distribution: sixth-order moments and cumulants



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Parameter estimation for INAR( ) processes


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LS estimation based on high-order statistics (HOS)

{x

1,x

2, . . . ,x

n}: realization of a non-negative integer-valued stationary stochastic

process with third-order moments (0, k), k> 0





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LS estimation based on high-order statistics (HOS)

{x

1,x

2, . . . ,x

n}: realization of a non-negative integer-valued stationary stochastic

process with third-order moments (0, k), k> 0

Approximating model: INAR(p) with parameters 1, ,p,e,2e andthird-order moments X(0, k), k> 0, which can be represented in the following

matrix form: 3,X = M3,X+eX(0)1p

X(0, 1)

X(0, 2)

..

.

X(0,p)

=

X(0, 0) X(1, 1) . . . X(p1,p1)X(0, 1) X(0, 0) . . . X(p

2,p

2)

..

....

. . ....

X(0,p1) X(0,p2) . . . X(0, 0)

1

2

..

.

p

+eX(0)

1

1

..

.

1

X(0) =p

i=1

iX(i) +eX + Vp, with Vp = e2 +X

p

i=1

i2





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Defining H = [M3,X X(0)1p] and = [ 1 p e ]T

3,X = H





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3,X = H

may be estimated by least squares, ie, minimizing the squared error between 3,Xand the third-order moments of the data:

3 = [ (0, 1) (0,p) ]T





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3,X = H


3 = [ (0, 1) (0,p) ]T

Least Squares estimator of using HOS (LS_HOS)

= min{L()} = min

{(3H)T(3H)}





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3,X = H


3 = [ (0, 1) (0,p) ]T

Least Squares estimator of using HOS (LS_HOS)

= min{L()} = min

{(3H)T(3H)}

In practice: = min{ L()} = min

{(3 H)T(3 H)}





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Spectral density function easy to obtain

{Xt} is an INAR process[Silva (2005)]

{Xt} is a Non-Gaussian Mixing process: {Xt} is strictly stationary, E[|Xt|k] < , t Z, k N,

s1=

. . .

sk1=

|CX(s1, . . . , sk1)| < , k 2, (mixing condition)





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s1=

. . .

sk1=


IN(j) =1

2N

N

t=1

Xteijt

2

f(j)2

22 ,





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s1=

. . .

sk1=


IN(j) =1

2N

N

t=1

Xteijt

2

f(j)2

22 ,

Whittle estimator [Silva and Oliveira (2004, 2005)]

= min{L(X)} = min

[N/2]

j=1

log(f(j)) +

IN(j)

f(j)





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(p) p




s1=

. . .

sk1=


IN(j) =1

2N

N

t=1

Xteijt

2

f(j)2

22 ,

Whittle estimator [Silva and Oliveira (2004, 2005)]

= min{L(X)} = min

[N/2]

j=1

log(f(j)) +

IN(j)

f(j)

Asymptotic variance: Fourth-order cumulant spectral density function





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(p) p

e

= X1p

i=1

i , 2e = VpXp

i=12

i,

where Vp = R(0)pi=1 i R(i) and 2i is an estimator of Var[Yj,i], i = 1, . . . ,p. Forinstance, 2i = i(1 i), for the binomial thinning operation.





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(p) p

e

= X1p

i=1

i , 2e = VpXp

i=12

i,


Simulation results

Non-admissible estimates constrained estimation (CLS, WHT, LS_HOS)





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( )

e = X1p

i=1i , 2e = VpX

p

i=12

i,


Simulation results

Non-admissible estimates constrained estimation (CLS, WHT, LS_HOS)Sample bias, variance, mean square error and univariate skewness decrease as

the sample size increases consistency and symmetry





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e = X1p

i=1i , 2e = VpX

p

i=12

i

,


Simulation results

Non-admissible estimates constrained estimation (CLS, WHT, LS_HOS)Sample bias, variance, mean square error and univariate skewness decrease as

the sample size increases consistency and symmetryLS_HOS, WHT and CLS provides good results

in terms of smaller sample bias, variance and mean square error





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Poliomyelitis incidence in the United States

Monthly number of U.S. cases of

poliomyelitis, from 1970 to 1983

[Zeger (1988): Parameter-driven

model], and sample

autocorrelation and partialautocorrelation functions

1970 1972 1974 1976 1978 1980 1982 19840

2

4

6

8

10

12

14

year

monthlycounts

0 5 10 15 20 25 300.2

0

0.2

0.4

0.6

0.8

1

1.2

k

(k)

0 5 10 15 20 25 300.2

0.1

0

0.1

0.2

0.3

k

(k)

Application to real data PDMA-UP - October 2009 22 / 30




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X= 1.33 and S2 = 3.48

INAR(1) with binomial thinning operation and discrete arrival process

Xt = Xt1 + et, t= 2, . . . , 168, E[et] = e, Var[et] = e2,





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X= 1.33 and S2 = 3.48

INAR(1) with binomial thinning operation and discrete arrival process

Xt = Xt1 + et, t= 2, . . . , 168, E[et] = e, Var[et] = e2,

Method e 2e

YW 0.2948 0.9403 2.9041

CLS 0.3063 0.9414 2.8862

WHT 0.2799 0.9601 2.9279

LS_HOS 0.2083 0.9277 3.0504

LS_HOS_C 0.2344 0.7477 3.0040

TOR_1B 0.1475 1.1367 3.1650

TOR_2B 0.1431 1.1425 3.1737

Table: Parameter estimates of the INAR(1) model fitted to the polio data.





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Number of plants within the industrial sector

The number of Swedish

mechanical paper and

pulp mills, from 1921 to

1981 [Brnns (1995)

and Brnns and

Hellstrm (2001):

Explanatory variables]

1920 1930 1940 1950 1960 1970 19815

10

15

20

25

30

35

40

45

50

Numberofplants





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Simple INAR(1)

It is not assumed the Poisson distribution for the arrival process:

X= 20.40 and S2 = 155.16





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Simple INAR(1)


X= 20.40 and S2 = 155.16

Method e 2e x

2x MSE

CLS 0.9591 0.2017 15.2268 4.9279 192.2764 9.3254LS_HOS 0.9269 1.3635 19.2253 18.6516 145.4513 9.2997

TOR_1B 0.9631 0.7518 14.7219 20.374 213.5073 8.9224





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Simple INAR(1)


X= 20.40 and S2 = 155.16

Method e 2e x

2x MSE

CLS 0.9591 0.2017 15.2268 4.9279 192.2764 9.3254LS_HOS 0.9269 1.3635 19.2253 18.6516 145.4513 9.2997

TOR_1B 0.9631 0.7518 14.7219 20.374 213.5073 8.9224

Mean and variance of the estimated models: x = e

1 and 2

x =(1 )(e+ 2e )

(1 )2(1 + )





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Simple INAR(1)


X= 20.40 and S2 = 155.16

Method e 2e x

2x MSE

CLS 0.9591 0.2017 15.2268 4.9279 192.2764 9.3254LS_HOS 0.9269 1.3635 19.2253 18.6516 145.4513 9.2997

TOR_1B 0.9631 0.7518 14.7219 20.374 213.5073 8.9224

Mean and variance of the estimated models: x = e

1 and 2

x =(1 )(e+ 2e )

(1 )2(1 + )

MSE between the observations and the fitted models based on TOR_1B, LS_HOS and

CLS estimates





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The number of plants and

the fitted values

considering the LS_HOS

and CLS estimates

1920 1930 1940 1950 1960 1970 19815

10

15

20

25

30

35

40

45

50

Numberofplants

Real data

CLS

LS_HOS



Recent developments


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Overdispersion new thinning operations and/or different distributions for thearrival processes

Extreme value theory

INAR with periodic structureOutliers

Forecasting

Heteroskedasticity

Random-coefficient INAR

Recent developments PDMA-UP - October 2009 27 / 30


Theme proposal


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Random-coefficient integer-valued autoregressive processes

Theme proposal PDMA-UP - October 2009 28 / 30


Theme proposal


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Random-coefficient integer-valued autoregressive processes

Describe the first-order random-coefficient integer-valued autoregressive,

RCINAR(1), process proposed by Gomes and Canto e Castro (2009) and Zheng et al.

(2007) and explain the principal differences/similarities between these two processes.

Gomes, D., Canto e Castro, L., 2009. Generalized integer-valued random coefficient for a first order

structure autoregressive (RCINAR) process. Journal of Statistical Planning and Inference, vol. 139

(12), pp. 40884097.

Zheng, H., Basawa, I. V., Datta, S., 2007. First-order random coefficient integer-valued

autoregressive processes. Journal of Statistical Planning and Inference, vol. 137 (1), pp. 212229.

Theme proposal PDMA-UP - October 2009 28 / 30


References I


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First-order integer-valued autoregressive (INAR(1))

process.

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Integer-valued moving average (INMA) process.

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Alzaid, A.A. and Al-Osh, M.A., 1990.

An integer-valued pth-order autoregressive structure

(INAR(p)) process.

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Brnns, K., 1994.

Estimation and testing in integer-valued AR(1) models.

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Brnns, K., 1995.

Explanatory variables in the AR(1) count data model.

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Estimation in integer-valued moving average models.

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17, pp. 277291.

Dion, J-P. and Gauthier, G. and Latour, A., 1995.

Branching processes with immigration and integer-valued

time series.

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Du, Jin-Guan and Li, Yuan, 1991.The integer-valued autoregressive (INAR(p)) model.


Franke, J. and Seligmann, T., 1993.

Conditional maximum likelihood estimates for INAR(1)

processes and their application to modelling epileptic

seizure counts.

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Maurice B. Priestley, Chapman & Hall, pp. 310330.

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Multivariate first order integer valued autoregressions.

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References PDMA-UP - October 2009 29 / 30


References II


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Gauthier, G. and Latour, A., 1994.

Convergence forte des estimateurs des paramtres dtun

processus GENAR(p).

Annales des Sciences Mathmatiques du Qubec, vol. 18,

pp. 4971

Latour, A., 1997.

The multivariate GINAR(p) process.

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Latour, A., 1998.

Existence and stochastic structure of a non-negative

integer-valued autoregressive process.


McKenzie, E., 1985.

Some simple models for discrete variate time series.

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McKenzie, E., 1986.

Autoregressive moving-average processes with

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McKenzie, E., 1988.

Some ARMA models for dependent sequences of Poisson

counts.

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Silva, M.E. and Oliveira, V.L., 2004.

Difference equations for the higher-order moments and

cumulants of the INAR(1) model.


Silva, M.E. and Oliveira, V.L., 2005.

Difference equations for the higher-order moments and

cumulants of the INAR(p) model.


Silva, I. e Silva, M.E., 2006.

Asymptotic distribution of the Yule-Walker estimator for

INAR(p) processes.

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Steutel, F.W. and Van Harn, K., 1979.

Discrete analogues of self-decomposability and stability.

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Steutel, F.W. and Vervaat, W. and Wolfe, S.J., 1983.

Integer valued branching processes with immigration.Advances in Applied Probability, vol. 15, pp. 713725.

Zeger, S.L., 1988.

A regression model for time series of counts.

Biometrika, vol. 75, pp. 621629.

References PDMA-UP - October 2009 30 / 30

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