p-ORDER ROUNDED INTEGER-VALUED AUTOREGRESSIVE

p-ORDER ROUNDED INTEGER-VALUED AUTOREGRESSIVE (RINAR(p)) PROCESS

By M. Kachour

arXiv:0902.1598v1 [stat.ME] 10 Feb 2009

IRMAR, University of Rennes 1, France.

M. Kachour IRMAR Institut de Recherche en Math´ematiques de Rennes UMR 6626/ Universit´e Rennes 1 campus scientifique de Beaulieu 263 Avenue de G´en´eral Leclerc 35042 RENNES CEDEX Email : [email protected] Tel : 00.33.(0)2.23.23.63.98

1

Abstract An extension of the RINAR(1) process for modelling discrete-time dependent counting processes is considered. The model RINAR(p) investigated here is a direct and natural extension of the real AR(p) model. Compared to classical INAR(p) models based on the thinning operator, the new models have several advantages: simple innovation structure ; autoregressive coefficients with arbitrary signs ; possible negative values for time series ; possible negative values for the autocorrelation function. The conditions for the stationarity and ergodicity, of the RINAR(p) model, are given. For parameter estimation, we consider the least squares estimator and we prove its consistency under suitable identifiability condition. Simulation experiments as well as analysis of real data sets are carried out to assess the performance of the model.

Keywords: Integer-valued time series, INAR models, rounding operator, RINAR(1) model, RINAR(p) model, least squares estimator. AMS classification code: Primary 62M 10, secondary 62M 20.

1

Introduction.

The integer-valued autoregressive, INAR(1), process is introduced by Al-osh & Alzaid [2] (1987) to model non-negative integer-valued phenomena that evolve in time. These models are based on the binomial thinning operator, denoted ◦, see [16]. The INAR(1) process is defined by Xt = a1 ◦ Xt−1 + εt , ∀ t ∈ N, where

(1)

Xt−1

a1 ◦ Xt−1 =

X

ξ1k .

k=1

Here, the so-called counting series (ξ1k ) are independent and identically distributed (i.i.d.) Bernoulli random variables with success probability a1 ∈ [0, 1] and (εt ) is a sequence of i.i.d. non-negative integer-valued random variables and independent of the counting series. Thus, a1 ◦ Xt−1 is a binomial random variable with a1 and Xt−1 as parameters, a1 ◦ Xt−1

B(Xt−1 , a1 ).

The INAR(p) is an analogue of equation (1) with p lags. An INAR(p) process is recursively defined by Xt = a1 ◦ Xt−1 + a2 ◦ Xt−2 + . . . + ap ◦ Xt−p + εt , ∀ t ∈ N, where, for i = 1, . . . , p, Xt−i

ai ◦ Xt−i =

X

k=1

2

ξik .

(2)

Here (ξ1k ), i ∈ {1, . . . , p} and k ∈ N \ {0} are independent Bernoulli-distributed variables, where ξ1k has success probability ai ∈ [0, 1] and (εt ) is a sequence of i.i.d. non-negative integer-valued random variables and independent of all the counting series. The general INAR(p) processes where first introduced by Al-osh & Alzaid [3] (1990) but Du & Li [7] (1991) proposed a different specification. In the specification of Du & Li [7] (1991), denoted INAR-DL, the autocorrelation structure of an INAR(p) process is the same as that of an AR(p) process, whereas it corresponds to the one of an ARMA(p,p-1) process in the specification of Al-osh & Alzaid [3] (1990), denoted INAR-AA. In particular, an INAR(2) process follows the equation

Xt = a1 ◦ Xt−1 + a2 ◦ Xt−2 + εt ,

(3)

where Xt−1

a1 ◦ Xt−1 =

X

ξ1k

B(Xt−1 , a1 )

(4)

ξ2k

B(Xt−2 , a2 ).

(5)

k=1

and Xt−2

a2 ◦ Xt−2 =

X

k=1

Note that each variable Xu is thinned twice : a1 ◦ Xu for Xu+1 and a2 ◦ Xu for Xu+2 . We distinguish two different models according to the construction of these thinning operations: The INAR(2)-AA specification introduced by Al-osh & Alzaid [3] (1990). Here, the counting series are chosen such that for each Xu , the vector (a1 ◦ Xu , a2 ◦ Xu , Xu − a1 ◦ Xu − a2 ◦ Xu ) follows a trinomial distribution with parameters (Xt−j ; a1 , a2 , 1 − a1 − a2 ). An important consequence arises from this choice of the counting series: a moving average structure is induced and the autocorrelation function of the process is similar to that of a ARMA(2,1) process. Du & Li [7] (1991) propose a modified specification, denoted INAR(2)-DL. The counting series are chosen such that {a1 ◦ Xu , a2 ◦ Xu , u ∈ Z} are independent. The correlation properties of this process are identical to the AR(2) model. Estimators of the parameters of INAR(p) are provided by several authors. For p = 1 and under the assumption that the sequence (εt ) has a Poisson distribution Franke & Seligmann [9] (1993) analysed maximum likelihood. Du & Li [7] (1991) derived the limit distribution of the OLS estimator of a = (a1 , . . . , ap ). Br¨ anna¨ as and Hellstr¨om [5] (2001) considered GMM estimation, Silva and Oliveira [15] (2005) proposed a frequency domain-based estimator of a. Drost & al. [6] (2008) provided an efficient estimator of the parameters, and in particular, showed that the INAR(p) model has the Local Asymptotic Normality property. Thus, the one-step ahead forecast, based on the conditional expectations, for the previous models is beset by the problem that forecast values obtained will be real rather than integer-valued in all but very rare cases. A mapping into the discrete support of the series is readily obtained by 3

applying Gaussian brackets (integer part of), or by rounding to the nearest integer; the latter will be employed along this paper. The class of INAR(p) models has several drawbacks. Their innovation structure is complex, depending not only on the noise process (εt ), but also on the counting variables. The autoregressive coefficients aj ’s are restricted to the interval [0, 1]. In the case of an INAR(1) model for example, this restriction excludes negative autocorrelations. We introduce the following model p X αj Xt−j + λi + εt , Xt = h

t ∈ N,

(6)

j=1

where h·i represents the rounding operator to the nearest integer, (εt ) is a sequence of centered i.i.d. integer-valued random variables, defined on a probability space (Ω, A, P), λ and (αj ) are real parameters. We call this model RINAR(p) (for rounded integer-valued autoregression). RINAR(p) has many advantages compared to the previous INAR models. Its innovation structure is simple, generated only by the noise (εt ). Its one-step ahead least squares predictor is given by

p X ˆ T +1 = E (XT +1 | Xs , s ≤ T ) = h αj Xt−j + λi, X

(7)

j=1

which is integer-valued by construction. We shall also see that the RINAR(p) model can produce autocorrelation functions as rich as those of real AR(p), including negatives autocorrelations. Moreover, by construction the RINAR(p) model can analyze a time series with negative values, a situation not covered by any INAR model. Lastly, it is worth mentioning that the h·i is in fact the natural operation often used in the collection of integer-valued series. In this paper, we study in details the RINAR(p) model. First in section 2, we give conditions ensuring the stationarity and the ergodicity of the model. Next in section 3, we introduce the least squares estimator for the estimation of the model parameters. This estimator is proved consistent under suitable conditions on the model. Because of the discontinuity of the rounding operator, particular care is needed for both the formulation of the model identifiability condition and the computation of the least squares estimator. A specific algorithm for the last problem is introduced in section 4. We then present a small simulation experiment in section 5 to assess the performance of the estimator. In section 6, we analyze a well-known time series with RINAR(p) models where classical integer-valued models are unsuccessful. Finally, section 7 collects the proofs of all theorical results.

1.1

Some notations

The following notations and properties will be used along this paper. First, let us define N = {x ∈ Z : x ≥ 0}, Z>0 = {x ∈ Z : x > 0} and Z− = {x ∈ Z : x ≤ 0}. Now, we introduce several useful properties of the rounding operator h·i. Note that hai is clearly defined anywhere, unless if a = k +

1 2

where k ∈ Z. By convention, we take hk + 12 i = k + 1, k ∈ N 4

and hk − 21 i = k − 1, k ∈ Z− . Note that a → hai is an odd function. Let {a} be the fractional part of a ∈ R, {a} ∈ [0; 1[. Here, the fractional part of a negative number is a positive as {a} = {−a} = {|a|}, for example {−1.23} = {1.23} = 0.23. Let s be the sign function defined by s(a) = 1 if a ≥ 0, and s(a) = −1 if a < 0. Then, for all a ∈ R, we have: a = hai + s(a) {a} − s(a)11{a}≥ 12 .

(8)

Let [a] be the integer part of a ∈ R, for example [2.8] = 2 and [−6.3] = −6. Then, for all a ∈ R, we have: a = [a] + s(a) {a} .

(9)

The following lemma will be useful and it is a direct consequence of equations (8) and (9). Lemma 1. Let x ∈ R and a, b ≥ 0. 1. |hxi − x| ≤ 12 . 2. |hxi| = h|x|i ≤ |x| + 21 . 3. ha + bi = c + h{a} + {b}i, where c = hai + hbi − 11{a}≥ 21 − 11{b}≥ 12 . 4. {a + b} = {{a} + {b}}. 5. hai = [a] + h{a}i.

2

Ergodicity and stationarity of the RINAR(p) process.

The study of the RINAR(p) process can be carried out though the following vectorized process 

   Yt =    

Xt Xt−1 .. . Xt−p+1



 P p h j=1 αj Xt−j + λi + εt     Xt−1   = ..     .   Xt−p+1



   .   

(10)

The process (Yt ) formes an homogeneous Markov chain with state space E = Zp and transition probability function p X αj xj + λi) 11y2 =x1 ,...,yp =xp−1 , ∀ x = (xj ), y = (yj ) ∈ E. π(x, y) = P(ε1 = y1 − h

(11)

j=1

The following proposition gives the conditions which ensure the ergodicity and the stationarity of the RINAR(p) process. For x = (x1 , . . . , xp ) ∈ Rp , let kxk1 = |x1 | + . . . + |xp |. For any measure R µ and function g on E, we set µ(g) = g(x)dµ(x).

Proposition 1. Suppose that:

5

1. The Markov chain (Yt ) is irreducible. 2. For some k > 1, E|εt |k < +∞. 3. Then

Pp

j=1

|αj | < 1.

1. The RINAR(p) process (Yt ) has an unique invariant probability measure µ which has a moment of order k (i.e. µ(k.kk1 ) < ∞). 2. For all y ∈ E and f ∈ L1 (µ) we have n

1X f (Yk ) −→ µ(f ), n

Py a.s.

k=1

where Py denotes the conditional probability P (. | Y0 = y).

3

Estimation of parameters

Let θ = (α1 , . . . , αp , λ) ∈ Rp+1 . In this section, it is assumed that θ belongs to a compact p

′

parameters space Θ, subset of ]−1, 1[ × R. Let x = (x1 , . . . , xp ) ∈ Rp . By convention, we round the vector x coordinate-wisely, i.e. hxi = (hx1 i, . . . , hxp i). We note p X αj xj + λi. f (x; θ) = f (x1 , . . . , xp ; θ) = h j=1

Then, the RINAR(p) model can be written in the following form     f (Yt−1 ; θ) εt         Xt−1    0      Yt =  ..  +  ..  = hM Yt−1 + ξi + ηt = F (Yt−1 ; θ) + ηt ,     .    .  Xt−p+1

0

where



M =

α1 . . . αp Ip−1

0



,



   ξ=   

λ 0 .. . 0





      

   and ηt =    

εt 0 .. . 0



   .   

Let X−P +1 , . . . , X0 , . . . , Xn be observations from the RINAR(p) process. For the estimation of the parameter θ, we consider the least squares estimator defined by θˆn := arg min ϕn (θ), θ∈Θ

where

n

ϕn (θ) =

(12)

n

1X 1X 2 (Xt − f (Yt−1 ; θ)) = (kYt − F (Yt−1 ; θ)k1 )2 . n t=1 n t=1 6

(13)

Some notations are necessary. To the norm k · k1 on Rp , we associate a norm on (Rp )2 by setting |z| := ky1 k1 + ky2 k1 for z := (y1 , y2 ) in (Rp )2 , where y1 and y2 are vectors in Rp . The actual value of the parameter is denoted θ0 = (α∗1 , . . . , α∗p , λ∗ ) and Pθ0 stands for the probability distribution of a.s.

the chain (Yt ) under the actual model. Moreover, any convergence −→ means an a.s. convergence under Pθ0 ,x , which hold independently of the initial state x. Let us make the following assumptions.

Assumption [H] 1. Under Pθ0 , the Markov chain (Yt ) is irreducible, 2. for some k ≥ 2, E|εt |k < +∞, 3.

Pp

j=1

|α∗j | < 1, where α∗j is the actual value of αj , p

4. the parametric space Θ, is a compact subset of ]−1, 1[ × R. Assume that [H] holds. Therefore, under Pθ0 and from Proposition 1, (Yt ) has an unique invariant measure µθ0 such that µθ0 (k.kk1 ) < ∞ for some k ≥ 2. Moreover, the double chain (Zt ) with Zt = (Yt , Yt−1 ) has similar properties. So, the chain (Zt ) has also an unique invariant measure denoted σθ0 . As µθ0 (k.kk1 ) < ∞, it follows that σθ0 (|.|k ) < ∞ for the |.| norm on E 2 defined above. Let the functions g(z; θ) = (ky − F (x; θ)k)2 , z = (x, y) ∈ E 2 , θ ∈ Θ,

(14)

K(θ) = σθ0 g(.; θ), θ ∈ Θ.

(15)

and

The following proposition give us the limit of the least squares estimating function where the convergence holds uniformaly on Θ. Proposition 2. Assume that [H] holds. Then, for any θ ∈ Θ, we have a.s.

1. ϕn (θ) −→ K(θ); 2

2. K(θ) − K(θ0 ) = µθ0 (f (.; θ) − f (.; θ0 )) . Moreover, a.s.

sup |ϕn (θ) − K(θ)| −→ 0. θ∈Θ

The proofs of this propositions and of all forthcoming results are postponed to section 7. Next, we consider the consistency problem of the least squares estimator. As in the RINAR(1) case ( see [12]), the identifiability of RINAR(p) model has a non-standard behavior. Because of rounding operations, the model identifiability depends whether the autoregressive coefficients α∗j ′

′

′

′

are rational or not. For two parameter vectors θ = (α1 , . . . , αp , λ), θ = (α1 , . . . , αp , λ ) of Θ, we define their distance to be n o ′ ′ ′ d(θ, θ ) = max |αj − αj |, 1 ≤ j ≤ p, |λ − λ | . 7

(16)

3.1

Strong consistency of the least squares estimator when at least one of αj∗ is irrational

The following proposition adresses precisely the question of identifiability of the parameters of RINAR(p) for the case where there exists at least j ∈ {1, . . . , p} such that α∗j ∈ R\Q. We recall, the following function p X αj xj + λi, ∀ x = (x1 , . . . , xp ) ∈ E and θ = (α1 , . . . , αp , λ) ∈ Θ. f (x; θ) = h j=1

Proposition 3. Assume that [H] holds. If there exists at least one j ∈ {1, . . . , p} such that α∗j is irrational. Then, f (x; θ) = f (x; θ0 ), ∀ x ∈ E ⇐⇒ θ = θ0 . Proposition 4. Assume that [H] holds. If there exists at least one j ∈ {1, . . . , p} such that α∗j is irrational. Then, for all (sufficiently small) ε > 0, we have inf |K(θ) − K(θ0 )| > 0,

θ∈Θε

where Θε = {θ : d(θ, θ0 ) ≥ ε}. Theorem 1. Assume that 1. there exists at least one j ∈ {1, . . . , p} such that α∗j is an irrational number; 2. under Pθ0 , the Markov chain (Yt ) is irreducible; 3. for some k ≥ 2, E|εt |k < +∞; 4.

Pp

j=1

|α∗j | < 1, where α∗j is the actual value of αj ; p

5. the parametric space Θ, is a compact subset of ]−1, 1[ × R. Therefore, the least squares estimators are strongly consistent estimators of the actual values of the parameters, i.e. θˆn → θ0 ,

3.2

Pθ0 − a.s.

Strong consistency of the least squares estimator when all the αj∗ are rationals

First, we recall the main result of the identifiability problem for RINAR(1) model [12], defined by Xt = hαXt−1 + λi + εt = f (Xt−1 ; θ) + εt . Let θ0 = (α0 , λ0 ) be the actual value of the model.

8

m q ,

If α0 =

where m ∈ Z, q ∈ Z>0 , m and q are taken to be coprime, then f (x; θ) = f (x; θ0 ), ∀ x ∈ E ⇐⇒ α = α0 and λ ∈ I0 .

I0 is a 1q -length interval or

1 2q -length

interval where λ0 ∈ I0 .

The length of I0 depends on the parity of q and the position of {λ0 } : • If q is even, then I0 is a q1 -length interval. • If q is odd, then we distinguish two sub-cases : h h h h 1 1 – If {λ0 } ∈ 0; 2q ∪ 2q−1 2q ; 1 , then I0 is a 2q -length interval. h h 1 2q−1 – If {λ0 } ∈ 2q ; 2q , then I0 is a 1q -length interval.

For the RINAR(p) model, we suppose that, for j = 1, . . . , p we have α∗j =

aj bj ,

where aj ∈ Z

∗

and bj ∈ N , aj and bj are taken to be coprime (aj ∧ bj = 1). Let θ0 = (α∗1 , . . . , α∗p , λ∗ ) be the actual value of the RINAR(p) model defined by (6). Next, we will show that RINAR(p) is equivalent to a RINAR(1) model with a rational parameter denoted ν0 . ′

Let y = (x1 , . . . , xp ) ∈ E. Thus, p X i=1

where

α∗i xi

= Qp

1

j=1 bj

p Y

Al = al

p X

Al xl

l=1

!

,

bj .

(17)

(18)

j=1,j6=l

From the B´ezout theorem, we get A1 Z + · · · + Ap Z = dZ,

(19)

where d = A1 ∧ · · · ∧ Ap is the P.G.C.D. of A1 , . . . , Ap . Then, there exists x ∈ Z such that p X

α∗i xi = ν0 x,

(20)

i=1

with ν0 = Qp

d

j=1 bj

∈ ]−1, 1[ .

(21)

We note that the numerator and denumerator of ν0 are not necessary coprime. Then, we rewrite ν0 with its irreductible fraction form ν0 = where a ∈ Z, b ∈ N∗ and a ∧ b = 1.

9

a , b

(22)

Proposition 5. Assume that [H] holds. Let θ = (α∗1 , . . . , α∗p , λ) ∈ Θ. It follows ∀ y ∈ E, f (y; θ) = f (y; θ0 ) ⇐⇒ ∀ x ∈ Z, hν0 x + λi = hν0 x + λ∗ i. Proposition 6. Assume that [H] holds. If for all j = 1, . . . , p we have α∗j =

aj bj ,

where aj ∈ Z and

bj ∈ N∗ , then f (x; θ) = f (x; θ0 ), ∀ x ∈ E ⇐⇒ αj = α∗j , ∀ j = 1, . . . , p and λ ∈ I0 . I0 is a 1b -length interval or

1 2b -length

interval where λ∗ ∈ I0 .

The length of I0 depends on the parity of b and the position of {λ∗ } : • If b is even, then I0 is a 1b -length interval. • If b is odd, then we distinguish two sub-cases :  1   2b−1  1 – If {λ∗ } ∈ 0; 2b -length interval. ∪ 2b ; 1 , then I0 is a 2b   1 2b−1 ; 2b , then I0 is a 1b -length interval. – If {λ∗ } ∈ 2b

Next, in the order to simplify the presentation, we give full details only in the case where b is even.
Let α0 = α∗1 , . . . , α∗p . From the above discussions, there exists k0 ∈ {0, 1, . . . , b − 1} such that 

 k0 k0 + 1 {λ } ∈ i0 = . , b b ∗

We define, I0 = {λ : hλi = hλ∗ i and {λ} ∈ i0 } and E0 = {α0 } × I0 .

(23)

For example, we take the RINAR(4) model with α1∗ =

3 , 25

α2∗ =

3 , 8

α3∗ =

1 5

α4∗ = −

1 4

and λ∗ = 2.5.

It is follows, from (18), that 4 Y

bj = 800,

A1 = 480,

A2 = 1500,

A3 = 800 and A4 = −1000.

j=1

So, we have d = A1 ∧ A2 ∧ A3 ∧ A4 = 20. Then, we get, from (21), that ν0 = fraction form ν0 =

1 = 0.025. 40

Therefore, b = 40 is even, we have   3 3 1 1 E0 = ( , , , − ) × [2.5, 2.525[ . 25 8 5 4

10

20 800

and its irreductible

Proposition 7. Assume that [H] holds. If α∗j =

aj bj ,

aj ∈ Z and bj ∈ N∗ , aj and bj are coprime

∀ j = 1, . . . , p and b the denumerator of (22) is even, then for all (sufficiently small) ε > 0, we have inf |K(θ) − K(θ0 )| > 0,

θ∈Θ0ε

where Θ0ε = {θ : d(θ, E0 ) ≥ ε}. Theorem 2. Assume that 1. α∗j =

aj bj ,

aj ∈ Z and bj ∈ N∗ , aj and bj are coprime ∀ j = 1, . . . , p,

2. b the denumerator of (22) is even. 3. under Pθ0 , the Markov chain (Yt ) is irreducible, 4. for some k ≥ 2, E|εt |k < +∞, 5.

Pp

j=1

|α∗j | < 1, where α∗j is the actual value of αj ,

6. the parametric space Θ, is a compact subset of ]−1, 1[p × R. ˆn converges ˆn is strongly consistent while λ Then we get d(θˆn , E0 ) → 0, Pθ0 − a.s. In other words, α to an interval of length

1 b

containing λ∗ .

In the case where b is odd, Theorem 2 still holds where I0 (therefore E0 ) is remplaced by the corresponding intervals as mentioned in Proposition 6. We know that, from [12], for RINAR(1) model with α0 = the length of interval I0 is equal to

1 b0 .

a0 b0

such that a0 and b0 are coprime,

There is a natural question :

can we reduce, with p > 1 parameters, the length of the interval I0 ? Let us consider the RINAR(2) model with α∗1 =

a1 b1

and α∗2 =

a2 . b2

Then, we get ν0 =

a d = b1 b2 b

where d = a1 b2 ∧ a2 b1

and a ∧ b = 1.

Let A be an interval in R, we note by |A| the length of A. We have |I0 | = 1b . Thus b/a (b1 b2 )

and a ∧ b = 1

then b/b1 b2 .

It follows that 1 ≤ |Il |. b1 b2 The RINAR(1) model with the parameter α0 = α∗1 (resp. α∗2 ) produces the interval noted A1 (resp. A2 ). Then, |A1 | =

1 b1

and |A2 | =

1 b2 .

Let B = A1 ∩ A2 . We have   1 1 . , I0 ⊆ B =⇒ |I0 | ≤ |B| ≤ min b1 b2 11

Finally, 1 ≤ |I0 | ≤ min b1 b2



1 1 , b1 b2



.

Therefore, by increasing the autoregression order p, we can actually expect a reduction of the interval I0 where the parameter λ remains unidentifiable.

4

A numerical method to calculate θˆn

To calculate the least squares estimator, we generalize the minimization algorithm proposed in [12]. As in RINAR(1) model, the initialization step is given by the Yule-Walker’s method. Therefore, the generalized algorithm continues through successive dichotomous search steps. First, we will explain, in a general way, the algorithm for ϕn with a scalar θ. The aim is to find θˆn = arg min ϕn (θ). θ∈[a,b]

Here, the initial search interval is [a, b] (i.e. lef t = a and right = b). The objective function ϕn is evaluated for every step of the search at three different points: θˆk and their middel left (mid− lef t) and middel right (mid − right) points. According to the minimum value of ϕn (θˆk ), ϕn (mid − lef t) and ϕn (mid − right), θˆk , lef t and right change their actual value (see Figure 1). For example, if ϕn (αk ) is the minimum then lef t takes the value of the mid − lef t and right takes the value of the mid − right. This process stops when range = |rigth − lef t| ≤ 0.001. The following pseudo-code defines the used dichotomous search of θ. l e f t 0.

2. For θ ∈ Γ3 , without loss of generality we assume that ν0 > 0, λ∗ ≥ 0. Note that d(θ, E0 ) = d(λ, I0 ) = inf |λ − λ∗ |. λ∗ ∈I0

We distinguish four cases depending on the position of the fractional part of λ∗ and λ :   (a) Case A : i0 ⊂ 0, 12 and λ ≥ I0 .   (b) Case B : i0 ⊂ 0, 12 and λ ≤ I0 .   (c) Case C : i0 ⊂ 12 , 1 and λ ≥ I0 .   (d) Case D : i0 ⊂ 21 , 1 and λ ≤ I0 .

  • Case A : i0 ⊂ 0, 12 ; λ ≥ I0 . For this case, it is simple to verified that     b k0 + 1 1 , k0 ∈ 0, 1, . . . , − 2 and {λ} ≥ . b ≥ 4, {λ} ∈ 0, 2 2 b Therefore, d(θ, E0 ) = inf {λ} − {λ∗ } ≥ ε.

(43)

λ∗ ∈i1

   . Let x0 = There exists n ∈ k0 + 1, . . . , 2b − 1 such that {λ} ∈ nb , n+1 b 0

We know that, from B´ezout theorem, for x0 used here there exits x = such that

p X

b 2

− n fixed.

(x01 , . . . , x0p )

∈E

α∗j x0j = ν0 x0 .

j=1

For all {λ∗ } ∈ i0 we have  p X 

j=1

and from (43) we get  p X Also, for x0 =

b 2



α∗j x0j

α∗j x0j

j=1

 

+ {λ∗ } =

  

+ {λ} =







1 x0 b

1 x0 b





+ {λ∗ }
0, uniformly on Γ3 = θ : θ ∈ Γ3 , ∀ j ∈ {1, . . . , p} |αj − α∗j | ≤ η ,

we have |K(θ) − K(θ0 )| ≥ e0 > 0.  ′′ ′ For θ ∈ Γ3 = Γ3 \ Γ3 = θ : θ ∈ Γ3 , ∀ j ∈ {1, . . . , p} |αj − α∗j | > η the same argument for Γ1 can be applied here and then there exists d0 > 0 such that inf′′ |K(θ) − K(θ0 )| ≥ d0 > 0. 

θ∈Γ3

Proof of Theorem 2 The conclusion θˆn → E0 , almost surely, results from Propositions 2 and 7 by the same arguments used in the proof of Theorem 1. 

26

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0 −0.05

0.00

0.05

0.10

0.15

0.20

0.25

0 0.20

0.30

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 −0.45

0.35

0.25

0.30

0.35

0.40

0.45

0.50

0.55

−0.40

−0.35

−0.30

−0.25

−0.20

−0.15

−0.10

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

ˆ (left to right, top to Figure 2: Histograms of the parameters estimates α ˆ j , j = 1, . . . , 4 and λ bottom). 500 replications of time series of length 500. Actual values of the parameters are θ0 = 3 3 1 −1 5 , 8 , 5 , 4 , 2 ). ( 25

27

7 6 5 4 furth 3 2 1 0

0

100

200

300

400

500

Time

Figure 3: Plot of the F¨ urth data.

Series furth

0.2

Partial ACF

0.4

−0.2

0.0

0.0

0.2

ACF

0.6

0.4

0.8

0.6

1.0

Series furth

0

10

20

30

40

0

Lag

5

10

15

20

25

Lag

Figure 4: Sample autocorrelation and partial autocorrrelation functions of F¨ urth data.

28

The forecast errors of the 105 last observations.

3

2

1

0

−1

−2

0

20

40

60

80

100

ˆ T +1 , 400 ≤ T ≤ 504, of the F¨ Figure 5: The forecast errors εˆT +1 = XT +1 − X urth data, from a RINAR(2) fit.

References [1] M. Al-Osh and A. A. Alzaid. Integer-valued moving average (INMA) process. Statist. Hefte, 29, 1988. [2] A. A. Alzaid and M. Al-Osh. First-order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal., 8(3):261–275, 1987. [3] A. A. Alzaid and M. Al-Osh. An integer-valued pth-order autoregressive structure (INAR(p)) process. J. Appl. Probab., 27(2):314–324, 1990. [4] Kurt Br¨ ann¨as and Andreia Hall. Estimation in integer-valued moving average models. Appl. Stoch. Models Bus. Ind., 17(3):277–291, 2001. [5] Kurt Br¨ ann¨as and J¨orgen Hellstr¨om. Generalized integer-valued autoregression. Econometric Rev., 20(4):425–443, 2001. [6] Feike C. Drost, Ramon van den Akker, and Bas J. M. Werker. Local asymptotic normality and efficient estimation for INAR(P ) models. J. Time Ser. Anal., 29(5):783–801, 2008. 29

[7] J.-G. Du and Y. Li. The integer-valued autoregressive (INAR(p)) model. J. Times Ser. Anal., 12:129–142, 1991. [8] M. Duflo. Random Iterative Models, volume 34 of Applications of Mathematics (New York). Springer-Verlag, Berlin, 1997. [9] J. Franke and T. Seligmann. Conditional maximum likelihood estimates for INAR(1) processes and their application to modelling epileptic seizure counts. In Developments in time series analysis, pages 310–330. London, 1993. [10] R. F¨ urth. Statistic und wahrscheinlichkeitsnachwirkung. Physikalische Zeitschrift, 19:421–426, 1918. [11] Robert C. Jung and A. R. Tremayne. Binomial thinning models for integer time series. Stat. Model., 6(2):81–96, 2006. [12] M. Kachour and J.F. Yao. First-order rounded integer-valued autoregressive (RINAR(1)) process. Preprint, University of Rennes 1, 2008. [13] E McKenzie. Some ARMA models for dependent sequences of Poisson counts. Adv. in Appl. Probab., 20:822–835, 1988. [14] T. M. Mills and E. Seneta. Goodness-of-fit for a branching process with immigration using sample partial autocorrelations. Stochastic Process. Appl., 33(1), 1989. [15] Maria Eduarda Silva and Vera L´ ucia Oliveira. Difference equations for the higher order moments and cumulants of the INAR(p) model. J. Time Ser. Anal., 26(1):17–36, 2005. [16] F. W. Steutel and K. van Harn. Discrete analogues of self-decomposability and stability. Ann. Probab., 7(5):893–899, 1979. [17] A. W. van der Vaart. Asymptotic Statistics. Cambridge University Press, 1998.

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