# Fractional approaches for the distribution of innovation sequence of INAR(1) processes

In this paper, we present a fractional decomposition of the probability generating function of the innovation process of the first-order non-negative integer-valued autoregressive [INAR(1)] process to obtain the corresponding probability mass function. We also provide a comprehensive review of integer-valued time series models, based on the concept of thinning operators, with geometric-type marginals. In particular, we develop four fractional approaches to obtain the distribution of innovation processes of the INAR(1) model and show that the distribution of the innovations sequence has geometric-type distribution. These approaches are discussed in detail and illustrated through a few examples. Finally, using the methods presented here, we develop four new first-order non-negative integer-valued autoregressive process for autocorrelated counts with overdispersion with known marginals, and derive some properties of these models.

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## 1 Introduction

Models for count time series with overdispersion, based on thinning operators, have been discussed by many authors. The vast majority of the proposed integer-valued autoregressive processes are based on geometric-type distribution. There is an exhaustive list of INAR(1) processes with geometric-type marginal distributions, geometric-type count variables or geometric-type innovations.

McKenzie (1986) and Al-Osh & Aly (1992)

proposed INAR(1) processes with geometric and negative binomial distributions as marginals. Similarly,

Alzaid & Al-Osh (1988) discussed the geometric INAR(1) process [GINAR(1)]. Ristić et al.  (2009) introduced the geometric first-order integer-valued autoregressive [NGINAR(1)] model with geometric marginal distribution. Integer-valued time series, with geometric marginal distribution, generated by mixtures of binomial and negative binomial thinning operators have been considered by Nastić & Ristić (2012), Nastić et al.  (2012) and Ristić & Nastić (2012). Recently, Nastić et al.  (2016a)

constructed a new stationary time series model with geometric marginals, based on thinning operator, which is a mixture of Bernoulli and geometric distributed random variables.

Of course, a simple approach is to change the distribution of innovations. In this context, Jazi et al.  (2012) proposed the geometric INAR(1) model with geometric innovations, while Bourguignon (2018) introduced a first-order non-negative integer-valued autoregressive process with zero-modified geometric innovations based on binomial thinning.

In context of dependent Bernoulli counting variables, Ristić et al.  (2013), Miletić Ilić (2016) and Nastić et al.  (2017) introduced an integer-valued time series model with geometric marginals based on dependent Bernoulli count variables. Recently, Miletić Ilić et al.  (2017) introduced an INAR(1) model based on a mixed dependent and independent count series with geometric marginals.

Nastić et al.  (2016b) introduced an -states random environment non-stationary INAR(1) which, by its different values, represents the marginals selection mechanism from a family of different geometric distributions. Borges et al.  (2016) and Borges et al.  (2017) introduced geometric first-order integer-valued autoregressive processes with geometric marginal distribution based on -binomial thinning operator and -geometric thinning operator, respectively.

Other works that have recently appeared in the literature dealing with geometric-type INAR(1) processes are those of Nastić (2012) (shifted geometric INAR(1) process), Barreto-Souza (2015) (INAR(1) process with zero-modified geometric marginals) and Yang et al.  (2016) (threshold INAR(1) process). Final mention should be made to the work of Ristić et al.  (2012), Popović et al.  (2016) and Popović (2016) proposed bivariate INAR(1) time series models with geometric marginals.

In many recent works, it is not clear how the distribution of the innovation processes was derived. In this context, motivated by Feller (see 2008, p. 276), we formulate here some procedures to obtain the probability mass function of the innovation process of an INAR(1) process by rewriting the probability generating function (pgf) of the innovation process as a quotient of two polynomial functions of enequal degrees. In particular, we present four fractional approaches to obtain the probability mass function of innovation processes of the INAR(1) model when the distribution of the innovations sequence has geometric-type distribution. Furthermore, we put forward four new first-order non-negative integer-valued autoregressive processes (Examples 6, 7, 8 and 9) with inflated-parameter Bernoulli and inflated-parameter geometric marginals (Kolev et al. , 2000).

The rest of this paper is organized as follows. In Section 2, we develop four fractional approaches to obtain the distribution of innovation processes of the INAR(1) model and show that the distribution of the innovations sequence has geometric-type distribution. In addition, some illustrative examples and new models are presented. Finally, some concluding remarks are made in Section 3.

## 2 Main results

Let , and denote the set of integers, positive integers and real numbers, respectively. All random variables will be defined on a common probability space  . A discrete-time stationary non-negative integer-valued stochastic process is said to be a first-order integer-valued autoregressive [INAR(1)] process if it satisfies the equation

 Xt=α⊙Xt−1+ϵt=Xt−1∑j=1Nj+ϵt,t∈Z,

where is a thinning operator, , is an innovation sequence of independent and identically distributed non-negative integer-valued random variables, not depending on past values of , mean

and variance

. It is also assumed that the variables that define are independent of the variables from which other values of the series are calculated, and are such that . Moreover, we assume that all variables defining the thinning operations are independent of the innovation sequence . The autocorrelation function of is of the same form as in the case of the usual AR(1) processes.

The stationary marginal distribution of can be determined from the equation

 φX(s)=φX(φN(s))⋅φϵ(s), (1)

where , and denote the pgf’s of , and , respectively. Equation (1) can be used to obtain the distribution of the innovations sequence if the marginal distribution of the observable INAR(1) process is known. By deriving the pgf’s in (1), and using the stationarity of , it is easily shown that the pgf of is given by

 φϵ(s)=φX(s)φX(φN(s)).

Next, we present some methods to obtain the probability mass function of the innovation sequence of an INAR(1) process when we can rewrite the pgf of the innovation process as a quotient of two polynomial functions of enequal degrees.

### 2.1 Method 1: Fractional decomposition of the innovation distribution

As in Feller (2008), let us assume that the pgf of the innovations process is a rational function given by

 φϵ(s)=Up(s)Vq(s),

where and are two polynomials of degrees and , respectively, and that the equation has distinct roots with . Then, we readily have that .

The probability mass function (pmf) of the innovation process can then be expressed (see Feller, 2008, p. 276) as

 Pr[ϵt=m]=q∑i=1ρism+1i, (2)

where . If is smaller in absolute value than all other roots, the above pmf can be approximated (see Feller, 2008, p. 277) by as . Assuming that , and using some well-known algebraic manipulations, it is possible to rewrite the pgf of the innovation process (see Feller, 2008, p. 277) as

 φϵ(s)=Ur(s)+Uk(s)Vq(s),k

Then, the results above can be applied to the rational function .

###### Remark 2.1.

Note that taking in (2), the innovation process is a mixture of geometric distributions with parameters expressed as

 Pr[ϵt=m]=q∑i=1ci(μi1+μi)m(11+μi), (4)

where . We can find many different INAR(1) models proposed in the literature whose innovation processes are given in (4). These models will be called fractional integer-valued autoregressive process and denoted here by FINAR(1) processes.

###### Remark 2.2.

Feller (2008) has mentioned in p. 276 that it is a hard work to find the exact mixture distribution in (4) and a simple approximation could be of practical interest in order to get satisfactory solutions for inferential problems. In fact, if we suppose that is smaller than all the others, then as increases, we see that can be approximated by the geometric distribution with parameter .

Given the marginal pmf of INAR(0,1) process, the next remark presents in detail the proof of a recursive formula to obtain the innovation pmf. This alternative recursive procedure could be of interest if the pmf of the thinning operator is available.

###### Remark 2.3.

(Alternative approach) The innovation pmf can be written by the recursive formula

 Pr[ϵt=l]=−1b0l−1∑i=l−qcibl−i,l=q+1,q+2,….
###### Proof.

Suppose

 φX(s)=Up(s)=p∑i=0aisiandφα⊙X(s)=Vp(s)=q∑j=0bjsj% withp

where ’s and ’s are some coefficients such that and . Then, we readily have

 φϵ(s)⋅q∑j=0bjsj=p∑i=0aisi,∀s. (5)

Now, let

 φϵ(s)=∞∑l=0cjsl,

where ’s are some coefficients such that . Then, upon substituting this in (5), we obtain

 ∞∑l=0cjsl⋅q∑j=0bjsj=p∑i=0aisi,∀s. (6)

Upon comparing coefficients of on both sides of (6), we obtain the following equations:

 c0b0 = a0, c0b1+c1b0 = a1, c0b2+c1b1+c2b0 = a2, ⋮ c0bp+c1bp−1+c2bp−2+⋯+cpb0 = ap,

which can be expressed equivalently as

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣b000⋯0b1b00⋯0b2b1b0⋯0⋮⋮⋮⋱⋮bpbp−1bp−2⋯b0⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣c0c1c2⋮cp⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣a0a1a2⋮ap⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.

This gives

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣c0c1c2⋮cp⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦=B−1p⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣a0a1a2⋮ap⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,whereBp=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣b000⋯0b1b00⋯0b2b1b0⋯0⋮⋮⋮⋱⋮bpbp−1bp−2⋯b0⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,

which is clearly invertible since . Next, upon comparing coefficients of on both sides of (6), we get the following equations:

 c0bp+1+c1bp+c2bp−1+⋯+cp+1b0 = 0, c0bp+2+c1bp+1+c2bp+⋯+cp+2b0 = 0, ⋮ c0bq+c1bq−1+c2bq−2+⋯+cqb0 = 0.

This system of equations can be readily rewritten as follows

 cp+1b0 = −(c0bp+1+c1bp+c2bp−1+⋯+cpb1), cp+1b1+cp+2b0 = −(c0bp+2+c1bp+1+c2bp+⋯+cpb2), ⋮ cp+1bq−p−1+⋯+cqb0 = −(c0bq+c1bq−1+c2bq−2+⋯+cpbq−p).

which can be expressed equivalently as

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣b000⋯0b1b00⋯0b2b1b0⋯0⋮⋮⋮⋱⋮bq−p−1bq−p−2bq−p−3⋯b0⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣cp+1cp+2cp+3⋮cq⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣bp+1bpbp−1⋯b1bp+2bp+1bp⋯b2bp+3bp+2bp+1⋯b3⋮⋮⋮⋱⋮bqbq−1bq−2⋯bq−p⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣c0c1c2⋮cp⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣cp+1cp+2cp+3⋮cq⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦=−B−1q−p−1B∗B−1p⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣a0a1a2⋮ap⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,whereB∗=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣bp+1bpbp−1⋯b1bp+2bp+1bp⋯b2bp+3bp+2bp+1⋯b3⋮⋮⋮⋱⋮bqbq−1bq−2⋯bq−p⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.

Next, upon comparing coefficients of on both sides of (6), we get the following equations

 c1bq+c2bq−1+c3bq−2+⋯+cq+1b0 = 0,(sq+1) c2bq+c3bq−1+c4bq−2+⋯+cq+2b0 = 0,(sq+2) c3bq+c4bq−1+c5bq−2+⋯+cq+3b0 = 0,(sq+3) ⋮

which readily yields the following solutions:

 cq+1 = −1b0q∑i=1cibq−i+1, cq+2 = −1b0q+1∑i=2cibq−i+2, cq+3 = −1b0q+2∑i=3cibq−i+3, ⋮

and in general, we have

 cl=−1b0l−1∑i=l−qcibl−i,l=q+1,q+2,…

###### Example 1 (Geometric INAR(1) process).

As in McKenzie (1985), let us consider the marginal distribution of to be a geometric distribution with parameter with pmf

 Pr[Xt=m]=(1−θ)mθ,m=0,1,2,…

The GINAR(1) process is based on the binomial thinning operator (Steutel and Van Harn 1979), , where the so-called counting series is a sequence of independent and identically distributed Bernoulli random variables with , and satisfies the equation

 Xt=α∘Xt−1+ϵt,t∈Z.

Thus, the pgf of the innovation processe can be written as

 φϵ(s)=φXt(s)φα∘Xt−1(s)=U(s)V(s),|s|<1/(1−θ),

where

 U(s)=θ+(1−θ)α−(1−θ)αsandV(s)=1−(1−θ)s.

Since we have here two linear functions with , upon using a simple algebraic manipulation suggested in (3), the pgf of the innovation process can be rewritten as

 φϵ(s)=U(s)V(s)=α+(1−α)θ1−(1−θ)s.

In this example, we have shown that the GINAR(1) process is a FINAR(1) process with the polynomial functions and having the same degrees . By using the rational decomposition in (2) for the fractional function , it can be seen that the innovation process follows a zero-inflated geometric distribution given by

 Pr[ϵt=m]=α1{0}(m)+(1−α)θ(1−θ)m,m=0,1,2,…
###### Example 2 (New geometric INAR(1) process).

In this example, we consider the stationary NGINAR(1) process with negative binomial thinning operator, introduced by Ristić et al.  (2009), given by

 Xt=α∗Xt−1+ϵt,t∈Z,

where , is a sequence of iid random variables with geometric distribution, i.e., with pmf given by , and is a stationary process with geometric marginals with pmf given by . Using these assumptions of the NGINAR(1) process, the pgf of the innovation sequence is given by

 φϵ(s)=U(s)V(s),|s|<1,

where and .

Since, NGINAR(1) process is a fractional INAR(1) process with and , the goal is to apply the fraction decomposition described before to obtain the pmf of the innovation processe . The decomposition proceeds as follows:

• The roots of the quadratic function are

 s1=1+μμands2=1+αα;
• The coefficients and are

 ρ1=−U(s1)V′(s1)=1μ(1−αμμ−α)andρ2=−U(s2)V′(s2)=μμ−α.

Thus, the pmf of the innovation sequence is obtained from (2) and is given by

 Pr[ϵt=m]=(1−αμμ−α)(μ1+μ)m11+μ+αμμ−α(α1+α)m11+α,m=0,1,2,…

The pmf of the innovation sequence is well defined with the condition for being necessary to guarantee that all probabilities are non-negative, and is undefined for values outside this range. Note that this result was obtained earlier in Ristić et al.  (2009). .

###### Example 3 (Dependent counting INAR(1) process).

Ristić et al.  (2013) proposed a geometrically distributed time series generated by dependent Bernoulli count series, called DCINAR(1) process. The DCINAR(1) process is based on new generalized binomial thinning operator and satisfies the following equation:

 Xt=α∘θXt−1+ϵt,t∈Z,

where the operator is defined as , , and is a sequence of Bernoulli() variables with and . Moreover, these random variables are dependent, since and . For more details, see Ristić et al.  (2013).

The pgf of the innovation sequence can be expressed into partial fractions as

 φϵ(s)=α(1−θ)(α+θ−αθ)α+θ−2αθ+ρ1s1−s+ρ2s2−s,|s|<1, (7)

where

 ρ1=A1μ, ρ2=A2(α+θ−2αθ)μands1=1+μμ, s2=1+(α+θ−2αθ)μα+θ−2αθ)μ.

The expressions for and are given in Ristić et al.  (2013). Taking and , the mixture innovation distribution in Ristić et al.  (2013) follows from (4) and (7) as

 Pr[ϵt=m]=A01{0}(m)+A1(μ11+μ1)m11+μ1+A2(μ21+μ2)m11+μ2, (8)

where

 A0=1−(A1+A2)=α(1−θ)(α+θ−αθ)α+θ−2αθ,A0+A1+A2=1andAi≥0,i=0,1,2.

We note that the probability distribution in (

8) is a zero-inflated model involving two geometric distributions. This result is not cleary stated in Theorem 2 of Ristić et al.  (2013).

### 2.2 Method 2: Linear rational probability generation function

In this section, we extend the innovation sequence of the geometric INAR(1) process discussed in Example 1. We consider here the a linear fractional probability generating function with four-parameters given by

 φϵ(s)=a+bsc+ds,s∈[0,1]andφϵ(0)<1, (9)

with the following parametric restrictions: and . After a simple algebraic manipulation, the pgf in (9) can be decomposed into partial fractions as

 φϵ(s) = bd+ρs1−s,

where and . Following the same procedure as in Feller (2008, p. 276), we obtain an exact expression for the pmf of the innovation sequence as

 Pr[ϵt=m]=bd1{0}(m)+ρsm+11,m=0,1,2,…

Taking , we have the pmf of the three-parameter innovation sequence

 Pr[ϵt=m]=bd1{0}(m)+(1−bd)(1−θ)mθ,m=0,1,2,… (10)

Note that the two-parameter innovation distribution in Example 1 is deduced by taking and . The mean and the variance of the innovation process in (10) are given by

 E[ϵt] = (1−θθ)(1−bd), Var[ϵt] = (1−θθ)(1−bd)[1θ(1+bd)−bd],

respectively.

The Fisher index of the dispersion of innovation process is given by

 Iϵ=Var[ϵt]E[ϵt]=1θ(1+bd)−bd.

This Fisher index indicates equidispersion of the innovation process if , underdispersion if and overdispersion if . Note that for , we have the mean to equal the variance, but the pmf in (10

) is not the classical Poisson distribution.

###### Example 4 (Two-parameter innovation sequence).

Let us now consider the following two-parameter pgf discussed by Aly & Bouzar (2017)

 φϵ(s)=1−m1−s1+r(1−s),0≤s≤1,

where and . It can be easily seen that the above pgf is a fraction linear function with and . For this particular case we have and . The corresponding two-parameter innovation process is given by

 Pr[ϵt=m]=(1−mr)1{0}(m)+(mr)(11+r)(r1+r)m,m=0,1,2,…
###### Example 5 (Zero-modified geometric innovation process).

Johnson et al.  (2005) have formulated in Section 8.2.3 the following pgf

 φϵ(s)=1−k+k11+μ(1−s),|s|<1, (11)

with the parametric restriction . It is straightforward to see that the pgf in (11) has a linear representation given by

 a=1+kμ,b=−kμ,c=1+μandd=−μ.

Note that and the parametric restriction implies that . Also, we have in view of (10), with , that the pmf of is given by

 Pr[ϵt=m]=k1{0}(m)+(1−k)(11+μ)(μ1+μ)mm=0,1,…

This pmf is the so-called zero-modified geometric distribution which presents equidispersion when ; underdispersion when and ; and overdispersion when . For , it is known as zero-inflated distribution and zero-deflated distribution for The zero-modified geometric distribution was considered in a recent paper by Bourguignon (2018) to formulate a new INAR(1) process with zero-modified geometric innovations. Also, if we take , the zero-modified geometric distribution is the Bernoulli-Geometric distribution with parameters and introduced by Bourguignon & Weiß (2017). This can be easily seen by taking the linear representation (see Bourguignon & Weiß, 2017)

 a=1−π,b=π,c=1+μandd=−μ.

### 2.3 Method 3: Quadratic rational probability generation function

In this section, we consider the pgf of the innovation process as a quadratic rational function expressed as follows:

 φϵ(s)=as2+bs+cas2+¯bs+¯c,s∈[0,1]andφϵ(0)<1, (12)

where and .

Initially, to facilitate the partial fraction decomposition, we have to reduce the degree of the polynomial function in the numerator of (12) by using a simple algebraic manipulation. Thus, we obtain a new expression for the pgf as

 φϵ(s) = 1+(b−¯b)s+c−¯cas2+¯bs+¯c=1+U1(s)(s−s1)(s−s2), (13)

with

 U1(s)=(b−¯b)(s−1)a,

where and are two different roots of the polynomial function of the denominator in (12) such that

 as2+¯bs+¯c=a(s−s1)(s−s2).

In order to obtain the innovation distribution, from here on, we assume that . The rational function in (13) can now be decomposed into partial fractions (see Feller, 2008) and the pgf of the innovation variable can be rewritten as

 φϵ(s) = 1+ρ1s1−s+ρ2s2−s,

where

 ρ1=−U1(s1)s1−s2andρ2=−U1(s2)s2−s1.

Using the same geometric expansion for and as in Feller (2008), the pmf of the innovation sequence is obtained as

 Pr[ϵt=m]={π,ifm=0,(1−π)[w1(1−p1)pm−11+w2(1−p2)pm−12],if m≥1, (14)

where , and . Note that is a mixture of two geometric distributions, where and are the weights with . If , then and if , then has a hurdle geometric distribution with parameter . Also, if , then and if , has a hurdle geometric distribution with parameter .

Since in (14) the zeros come from the Bernoulli process with parameter and the nonzeros come from a different process characterized by a mixture of two geometric distributions, it may be referred to as “hurdle-fractional geometric innovation distribution” and denoted by HFG. The mean and the variance of the innovation process in (14) are given by

 E[ϵt] = (1−π)(1−p1)(1−p2), Var[ϵt] = (1−π)[p21p2−3p1p2+p1+p2+π](1−p1)2(1−p2)2,

respectively.

###### Example 6 (Inflated-parameter geometric INAR(1) process based on binomial thinning operator).

In this example, we present a new stationary first-order non-negative integer valued autoregressive process, , with inflated-parameter geometric (Kolev et al. , 2000) marginals. The proposed process is based on the binomial thinning operator and satisfies the equation:

 Xt=α∘Xt−1+ϵt,t∈Z,

with and being a stationary process with inflated-parameter geometric marginals, i.e., with pmf given by

 Pr[Xt=m]=ρμ+ρ1{0}(m)+(1−ρμ+ρ)(μ+ρ1+μ)m(1−ρ1+μ),m=0,1,2,…,

where and . The inflated-parameter geometric distribution is well-known as -geometric distribution (see Kolev et al. , 2000) and considered in a recent paper by Borges et al.  (2017) to formulate a new thinning operator. The corresponding pgf is given by

 φX(s)=1−ρs1−sρ+μ(1−s),|s|<1.

The mean and variance of the process are given, respectively, by

 E[Xt]=μ1−ρandVar[