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

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 $r$ -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 $ρ$ -binomial thinning operator and $ρ$ -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 $Z$ , $N$ and $R$ denote the set of integers, positive integers and real numbers, respectively. All random variables will be defined on a common probability space $(Ω, A, P)$ . A discrete-time stationary non-negative integer-valued stochastic process ${X_{t}}_{t \in Z}$ is said to be a first-order integer-valued autoregressive [INAR(1)] process if it satisfies the equation

X_{t} = α ⊙ X_{t - 1} + ϵ_{t} = X_{t - 1} \sum j = 1 N_{j} + ϵ_{t}, t \in Z,

where $α ⊙$ is a thinning operator, $α \in (0, 1)$ , ${ϵ_{t}}_{t \in Z}$ is an innovation sequence of independent and identically distributed non-negative integer-valued random variables, not depending on past values of ${X_{t}}_{t \in Z}$ , mean $μ_{ϵ} (< \infty)$

and variance

σ_{ϵ}^{2} (< \infty)

. It is also assumed that the

N_{j}

variables that define

α ⊙ X_{t - 1}

are independent of the variables from which other values of the series are calculated, and are such that

E (N_{j}) = α

. Moreover, we assume that all

N_{j}

variables defining the thinning operations are independent of the innovation sequence

{ϵ_{t}}_{t \in Z}

. The autocorrelation function of

{X_{t}}_{t \in Z}

is of the same form as in the case of the usual AR(1) processes.

The stationary marginal distribution of ${X_{t}}_{t \in Z}$ can be determined from the equation

φ_{X} (s) = φ_{X} (φ_{N} (s)) \cdot φ_{ϵ} (s),

(1)

where $φ_{X} (s) := E (s^{X})$ , $φ_{N} (s) := E (s^{N})$ and $φ_{ϵ} (s) := E (s^{ϵ})$ denote the pgf’s of ${X_{t}}_{t \in Z}$ , ${N_{j}}_{j = 1}^{X_{t - 1}}$ and ${ϵ_{t}}_{t \in Z}$ , 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 ${X_{t}}_{t \in Z}$ , it is easily shown that the pgf of ${ϵ_{t}}_{t \in Z}$ is given by

φ_{ϵ} (s) = \frac{φ_{X} (s)}{φ_{X} (φ_{N} (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

φ_{ϵ} (s) = \frac{U_{p} (s)}{V_{q} (s)},

where $U_{p} (s)$ and $V_{q} (s)$ are two polynomials of degrees $p$ and $q (p < q)$ , respectively, and that the equation $V_{q} (s) = 0$ has $q$ distinct roots $s_{1}, s_{2}, \dots, s_{q}$ with $| s | < min {s_{1}, s_{2}, \dots, s_{q}}$ . Then, we readily have that $V_{q} (s) = \prod_{i = 1}^{q} (s - s_{i})$ .

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

Pr [ϵ_{t} = m] = q \sum i = 1 \frac{ρ_{i}}{s_{i}^{m + 1}},

(2)

where $ρ_{i} = \frac{- U_{p} (s_{i})}{\frac{d V_{q} (s)}{d s} ∣_{s = s_{i}}}$ . If $s_{1}$ is smaller in absolute value than all other roots, the above pmf can be approximated (see Feller, 2008, p. 277) by $\frac{ρ_{1}}{s_{1}^{m + 1}}$ as $m ⟶ \infty$ . Assuming that $p = q + r, r \geq 0$ , and using some well-known algebraic manipulations, it is possible to rewrite the pgf of the innovation process (see Feller, 2008, p. 277) as

φ_{ϵ} (s) = U_{r} (s) + \frac{U_{k} (s)}{V_{q} (s)}, k < q .

(3)

Then, the results above can be applied to the rational function $U_{k} (s) / V_{q} (s)$ .

Remark 2.1.

Note that taking $s_{i} = (1 + μ_{i}) / u_{i}$ in (2), the innovation process is a mixture of geometric distributions with parameters $μ_{i} / (1 + μ_{i})$ expressed as

Pr [ϵ_{t} = m] = q \sum i = 1 c_{i} {(\frac{μ_{i}}{1 + μ_{i}})}^{m} (\frac{1}{1 + μ_{i}}),

(4)

where $c_{i} = ρ_{i} μ_{i}, i = 1, \dots, q$ . 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 $μ_{1}$ is smaller than all the others, then as $m$ increases, we see that $Pr [ϵ_{t} = m]$ can be approximated by the geometric distribution with parameter $μ_{1} / (1 + μ_{1})$ .

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 [ϵ_{t} = l] = - \frac{1}{b_{0}} l - 1 \sum i = l - q c_{i} b_{l - i}, l = q + 1, q + 2, \dots .

Proof.

Suppose

φ_{X} (s) = U_{p} (s) = p \sum i = 0 a_{i} s^{i} and φ_{α ⊙ X} (s) = V_{p} (s) = q \sum j = 0 b_{j} s^{j} % with p < q,

where $a_{i}$ ’s and $b_{i}$ ’s are some coefficients such that $a_{0} + a_{1} + \dots + a_{p} = 1$ and $b_{0} + b_{1} + \dots + b_{q} = 1$ . Then, we readily have

φ_{ϵ} (s) \cdot q \sum j = 0 b_{j} s^{j} = p \sum i = 0 a_{i} s^{i}, \forall s .

(5)

Now, let

φ_{ϵ} (s) = \infty \sum l = 0 c_{j} s^{l},

where $c_{l}$ ’s are some coefficients such that $\infty \sum l = 0 c_{l} = 1$ . Then, upon substituting this in (5), we obtain

\infty \sum l = 0 c_{j} s^{l} \cdot q \sum j = 0 b_{j} s^{j} = p \sum i = 0 a_{i} s^{i}, \forall s .

(6)

Upon comparing coefficients of $s^{0}, s^{1}, \dots, s^{p}$ on both sides of (6), we obtain the following equations:

$c_{0} b_{0}$	$=$	$a_{0},$
$c_{0} b_{1} + c_{1} b_{0}$	$=$	$a_{1},$
$c_{0} b_{2} + c_{1} b_{1} + c_{2} b_{0}$	$=$	$a_{2},$
	$⋮$
$c_{0} b_{p} + c_{1} b_{p - 1} + c_{2} b_{p - 2} + \dots + c_{p} b_{0}$	$=$	$a_{p},$

which can be expressed equivalently as

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} b_{0} & 0 & 0 & \dots & 0 b_{1} & b_{0} & 0 & \dots & 0 b_{2} & b_{1} & b_{0} & \dots & 0 ⋮ & ⋮ & ⋮ & ⋱ & ⋮ b_{p} & b_{p - 1} & b_{p - 2} & \dots & b_{0} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} c_{0} c_{1} c_{2} ⋮ c_{p} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{0} a_{1} a_{2} ⋮ a_{p} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

This gives

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} c_{0} c_{1} c_{2} ⋮ c_{p} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = B_{p}^{- 1} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{0} a_{1} a_{2} ⋮ a_{p} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦, where B_{p} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} b_{0} & 0 & 0 & \dots & 0 b_{1} & b_{0} & 0 & \dots & 0 b_{2} & b_{1} & b_{0} & \dots & 0 ⋮ & ⋮ & ⋮ & ⋱ & ⋮ b_{p} & b_{p - 1} & b_{p - 2} & \dots & b_{0} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦,

which is clearly invertible since $b_{0} \neq 0 (b_{0} = V_{q} (0) > 0)$ . Next, upon comparing coefficients of $s^{p + 1}, s^{p + 2}, \dots, s^{q}$ on both sides of (6), we get the following equations:

$c_{0} b_{p + 1} + c_{1} b_{p} + c_{2} b_{p - 1} + \dots + c_{p + 1} b_{0}$	$=$	$0,$
$c_{0} b_{p + 2} + c_{1} b_{p + 1} + c_{2} b_{p} + \dots + c_{p + 2} b_{0}$	$=$	$0,$
	$⋮$
$c_{0} b_{q} + c_{1} b_{q - 1} + c_{2} b_{q - 2} + \dots + c_{q} b_{0}$	$=$	$0.$

This system of equations can be readily rewritten as follows

$c_{p + 1} b_{0}$	$=$	$- (c_{0} b_{p + 1} + c_{1} b_{p} + c_{2} b_{p - 1} + \dots + c_{p} b_{1}),$
$c_{p + 1} b_{1} + c_{p + 2} b_{0}$	$=$	$- (c_{0} b_{p + 2} + c_{1} b_{p + 1} + c_{2} b_{p} + \dots + c_{p} b_{2}),$
	$⋮$
$c_{p + 1} b_{q - p - 1} + \dots + c_{q} b_{0}$	$=$	$- (c_{0} b_{q} + c_{1} b_{q - 1} + c_{2} b_{q - 2} + \dots + c_{p} b_{q - p}) .$

which can be expressed equivalently as

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} b_{0} & 0 & 0 & \dots & 0 b_{1} & b_{0} & 0 & \dots & 0 b_{2} & b_{1} & b_{0} & \dots & 0 ⋮ & ⋮ & ⋮ & ⋱ & ⋮ b_{q - p - 1} & b_{q - p - 2} & b_{q - p - 3} & \dots & b_{0} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} c_{p + 1} c_{p + 2} c_{p + 3} ⋮ c_{q} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} b_{p + 1} & b_{p} & b_{p - 1} & \dots & b_{1} b_{p + 2} & b_{p + 1} & b_{p} & \dots & b_{2} b_{p + 3} & b_{p + 2} & b_{p + 1} & \dots & b_{3} ⋮ & ⋮ & ⋮ & ⋱ & ⋮ b_{q} & b_{q - 1} & b_{q - 2} & \dots & b_{q - p} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} c_{0} c_{1} c_{2} ⋮ c_{p} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

This readily yields

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} c_{p + 1} c_{p + 2} c_{p + 3} ⋮ c_{q} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = - B_{q - p - 1}^{- 1} B^{*} B_{p}^{- 1} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{0} a_{1} a_{2} ⋮ a_{p} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦, where B^{*} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} b_{p + 1} & b_{p} & b_{p - 1} & \dots & b_{1} b_{p + 2} & b_{p + 1} & b_{p} & \dots & b_{2} b_{p + 3} & b_{p + 2} & b_{p + 1} & \dots & b_{3} ⋮ & ⋮ & ⋮ & ⋱ & ⋮ b_{q} & b_{q - 1} & b_{q - 2} & \dots & b_{q - p} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

Next, upon comparing coefficients of $s^{q + 1}, s^{q + 2}, \dots, .$ on both sides of (6), we get the following equations

$c_{1} b_{q} + c_{2} b_{q - 1} + c_{3} b_{q - 2} + \dots + c_{q + 1} b_{0}$	$=$	$0, (s^{q + 1})$
$c_{2} b_{q} + c_{3} b_{q - 1} + c_{4} b_{q - 2} + \dots + c_{q + 2} b_{0}$	$=$	$0, (s^{q + 2})$
$c_{3} b_{q} + c_{4} b_{q - 1} + c_{5} b_{q - 2} + \dots + c_{q + 3} b_{0}$	$=$	$0, (s^{q + 3})$
	$⋮$

which readily yields the following solutions:

$c_{q + 1}$	$=$	$- \frac{1}{b_{0}} q \sum i = 1 c_{i} b_{q - i + 1},$
$c_{q + 2}$	$=$	$- \frac{1}{b_{0}} q + 1 \sum i = 2 c_{i} b_{q - i + 2},$
$c_{q + 3}$	$=$	$- \frac{1}{b_{0}} q + 2 \sum i = 3 c_{i} b_{q - i + 3},$
	$⋮$

and in general, we have

c_{l} = - \frac{1}{b_{0}} l - 1 \sum i = l - q c_{i} b_{l - i}, l = q + 1, q + 2, \dots

∎

Example 1 (Geometric INAR(1) process).

As in McKenzie (1985), let us consider the marginal distribution of ${X_{t}}_{t \in Z}$ to be a geometric distribution with parameter $θ \in (0, 1)$ with pmf

Pr [X_{t} = m] = (1 - θ)^{m} θ, m = 0, 1, 2, \dots

The GINAR(1) process is based on the binomial thinning operator (Steutel and Van Harn 1979), $α \circ X_{t - 1} := \sum_{j = 1}^{X_{t - 1}} Y_{j}$ , where the so-called counting series ${Y_{j}}_{j \geq 1}$ is a sequence of independent and identically distributed Bernoulli random variables with $Pr (Y_{j} = 1) = 1 - Pr (Y_{j} = 0) = α \in [0, 1)$ , and satisfies the equation

X_{t} = α \circ X_{t - 1} + ϵ_{t}, t \in Z .

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

φ_{ϵ} (s) = \frac{φ_{X_{t}} (s)}{φ_{α \circ X_{t - 1}} (s)} = \frac{U (s)}{V (s)}, | s | < 1 / (1 - θ),

where

U (s) = θ + (1 - θ) α - (1 - θ) α s and V (s) = 1 - (1 - θ) s .

Since we have here two linear functions with $p = q = 1$ , upon using a simple algebraic manipulation suggested in (3), the pgf of the innovation process can be rewritten as

φ_{ϵ} (s) = \frac{U (s)}{V (s)} = α + (1 - α) \frac{θ}{1 - (1 - θ) s} .

In this example, we have shown that the GINAR(1) process is a FINAR(1) process with the polynomial functions $U (s)$ and $V (s)$ having the same degrees $p = q = 1$ . By using the rational decomposition in (2) for the fractional function $\frac{θ (1 - α)}{1 - (1 - θ) s}$ , it can be seen that the innovation process follows a zero-inflated geometric distribution given by

Pr [ϵ_{t} = m] = α 1_{{0}} (m) + (1 - α) θ (1 - θ)^{m}, m = 0, 1, 2, \dots

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

X_{t} = α * X_{t - 1} + ϵ_{t}, t \in Z,

where $α * X_{t - 1} = \sum_{i = 1}^{X_{t - 1}} W_{i}, α \in [0, 1)$ , ${W_{i}}_{t \in Z}$ is a sequence of iid random variables with geometric distribution, i.e., with pmf given by $Pr [W_{i} = m] = {(\frac{α}{1 + α})}^{m} \frac{1}{1 + α}$ , and ${X_{t}}_{t \in Z}$ is a stationary process with geometric $(μ / (1 + μ))$ marginals with pmf given by $Pr [X_{t} = m] = (\frac{μ}{1 + μ})^{m} \frac{1}{1 + μ}, μ > 0$ . Using these assumptions of the NGINAR(1) process, the pgf of the innovation sequence is given by

φ_{ϵ} (s) = \frac{U (s)}{V (s)}, | s | < 1,

where $U (s) = 1 + α (1 + μ) - α (1 + μ) s$ and $V (s) = α μ (s - \frac{1 + u}{μ}) (s - \frac{1 + α}{α})$ .

Since, NGINAR(1) process is a fractional INAR(1) process with $p = 1$ and $q = 2$ , the goal is to apply the fraction decomposition described before to obtain the pmf of the innovation processe ${ϵ_{t}}_{t \in Z}$ . The decomposition proceeds as follows:

The roots of the quadratic function $V (s)$ are

$s_{1} = \frac{1 + μ}{μ} and s_{2} = \frac{1 + α}{α};$
The coefficients $ρ_{1}$ and $ρ_{2}$ are

$ρ_{1} = - \frac{U (s_{1})}{V^{'} (s_{1})} = \frac{1}{μ} (1 - \frac{α μ}{μ - α}) and ρ_{2} = - \frac{U (s_{2})}{V^{'} (s_{2})} = \frac{μ}{μ - α} .$

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

Pr [ϵ_{t} = m] = (1 - \frac{α μ}{μ - α}) {(\frac{μ}{1 + μ})}^{m} \frac{1}{1 + μ} + \frac{α μ}{μ - α} {(\frac{α}{1 + α})}^{m} \frac{1}{1 + α}, m = 0, 1, 2, \dots

The pmf of the innovation sequence is well defined with the condition $α \in [0, μ / (1 + μ)]$ for $μ > 0$ 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 $α \circ_{θ}$ and satisfies the following equation:

X_{t} = α \circ_{θ} X_{t - 1} + ϵ_{t}, t \in Z,

where the operator $α \circ_{θ}$ is defined as $α \circ_{θ} X = \sum_{i = 1}^{X_{t - 1}} U_{i}$ , $α, θ \in [0, 1]$ , and ${U_{i}}_{i \in N}$ is a sequence of Bernoulli( $α$ ) variables with $E [U_{i}] = α$ and $Var [U_{i}] = α (1 - α)$ . Moreover, these random variables are dependent, since $Corr [U_{i}, U_{j}] = θ^{2}, \forall θ \neq 0$ and $i \neq j$ . For more details, see Ristić et al. (2013).

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

φ_{ϵ} (s) = \frac{α (1 - θ) (α + θ - α θ)}{α + θ - 2 α θ} + \frac{ρ_{1}}{s_{1} - s} + \frac{ρ_{2}}{s_{2} - s}, | s | < 1,

(7)

where

ρ_{1} = \frac{A_{1}}{μ}, ρ_{2} = \frac{A_{2}}{(α + θ - 2 α θ) μ} and s_{1} = \frac{1 + μ}{μ}, s_{2} = \frac{1 + (α + θ - 2 α θ) μ}{α + θ - 2 α θ) μ} .

The expressions for $A_{1}$ and $A_{2}$ are given in Ristić et al. (2013). Taking $μ_{1} = μ$ and $μ_{2} = (α + θ - 2 α θ) μ$ , the mixture innovation distribution in Ristić et al. (2013) follows from (4) and (7) as

Pr [ϵ_{t} = m] = A_{0} 1_{{0}} (m) + A_{1} {(\frac{μ_{1}}{1 + μ_{1}})}^{m} \frac{1}{1 + μ_{1}} + A_{2} {(\frac{μ_{2}}{1 + μ_{2}})}^{m} \frac{1}{1 + μ_{2}},

(8)

where

A_{0} = 1 - (A_{1} + A_{2}) = \frac{α (1 - θ) (α + θ - α θ)}{α + θ - 2 α θ}, A_{0} + A_{1} + A_{2} = 1 and A_{i} \geq 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

φ_{ϵ} (s) = \frac{a + b s}{c + d s}, s \in [0, 1] and φ_{ϵ} (0) < 1,

(9)

with the following parametric restrictions: $a < c$ and $a + b = c + d$ . After a simple algebraic manipulation, the pgf in (9) can be decomposed into partial fractions as

φ_{ϵ} (s)

=

\frac{b}{d} + \frac{ρ}{s_{1} - s},

where $s_{1} = - c / d$ and $ρ = b c / d^{2} - a / d$ . Following the same procedure as in Feller (2008, p. 276), we obtain an exact expression for the pmf of the innovation sequence as

Pr [ϵ_{t} = m] = \frac{b}{d} 1_{{0}} (m) + \frac{ρ}{s_{1}^{m + 1}}, m = 0, 1, 2, \dots

Taking $s_{1} = 1 / (1 - θ), 0 \leq θ < 1$ , we have the pmf of the three-parameter innovation sequence

Pr [ϵ_{t} = m] = \frac{b}{d} 1_{{0}} (m) + (1 - \frac{b}{d}) (1 - θ)^{m} θ, m = 0, 1, 2, \dots

(10)

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

	$E [ϵ_{t}]$	$=$	$(\frac{1 - θ}{θ}) (1 - \frac{b}{d}),$
	$Var [ϵ_{t}]$	$=$	$(\frac{1 - θ}{θ}) (1 - \frac{b}{d}) [\frac{1}{θ} (1 + \frac{b}{d}) - \frac{b}{d}],$

respectively.

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

I_{ϵ} = \frac{Var [ϵ_{t}]}{E [ϵ_{t}]} = \frac{1}{θ} (1 + \frac{b}{d}) - \frac{b}{d} .

This Fisher index indicates equidispersion of the innovation process if $b = - d$ , underdispersion if $b > - d$ and overdispersion if $b < - d$ . Note that for $b = - d$ , 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)

φ_{ϵ} (s) = 1 - m \frac{1 - s}{1 + r (1 - s)}, 0 \leq s \leq 1,

where $r \geq 0$ and $m \leq r + 1$ . It can be easily seen that the above pgf is a fraction linear function with $a = 1 - r - m, b = m - r, c = 1 + r$ and $d = - r$ . For this particular case we have $s_{1} = (r + 1) / r$ and $b / d = 1 - m / r$ . The corresponding two-parameter innovation process is given by

Pr [ϵ_{t} = m] = (1 - \frac{m}{r}) 1_{{0}} (m) + (\frac{m}{r}) (\frac{1}{1 + r}) {(\frac{r}{1 + r})}^{m}, m = 0, 1, 2, \dots

Example 5 (Zero-modified geometric innovation process).

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

φ_{ϵ} (s) = 1 - k + k \frac{1}{1 + μ (1 - s)}, | s | < 1,

(11)

with the parametric restriction $- 1 / μ \leq k \leq 1$ . It is straightforward to see that the pgf in (11) has a linear representation given by

a = 1 + k μ, b = - k μ, c = 1 + μ and d = - μ .

Note that $s_{1} = - c / d = (1 + μ) / μ$ and the parametric restriction $0 \leq a \leq c$ implies that $- 1 / μ \leq k \leq 1$ . Also, we have $b / d = k$ in view of (10), with $θ = 1 / (1 + μ)$ , that the pmf of ${ϵ_{t}}_{t \in Z}$ is given by

Pr [ϵ_{t} = m] = k 1_{{0}} (m) + (1 - k) (\frac{1}{1 + μ}) {(\frac{μ}{1 + μ})}^{m} m = 0, 1, \dots

This pmf is the so-called zero-modified geometric distribution which presents equidispersion when $k = - 1$ ; underdispersion when $μ \in (0, 1)$ and $k \in [- 1 / μ, - 1)$ ; and overdispersion when $k \in (- 1, 1)$ . For $0 < k < 1$ , it is known as zero-inflated distribution and zero-deflated distribution for $- 1 / μ < k < 0.$ 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 $k = - π / μ, 0 \leq π \leq 1$ , the zero-modified geometric distribution is the Bernoulli-Geometric distribution with parameters $μ$ and $π$ 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 + μ and d = - 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:

φ_{ϵ} (s) = \frac{a s^{2} + b s + c}{a s^{2} + ¯ b s + ¯ c}, s \in [0, 1] and φ_{ϵ} (0) < 1,

(12)

where $a \neq 0, b + c = ¯ b + ¯ c$ and $c \leq ¯ c$ .

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

φ_{ϵ} (s)

=

1 + \frac{(b - ¯ b) s + c - ¯ c}{a s^{2} + ¯ b s + ¯ c} = 1 + \frac{}{U_{1} (s)} (s - s_{1}) (s - s_{2}),

(13)

with

U_{1} (s) = \frac{(b - ¯ b) (s - 1)}{a},

where $s_{1}$ and $s_{2}$ are two different roots of the polynomial function of the denominator in (12) such that

a s^{2} + ¯ b s + ¯ c = a (s - s_{1}) (s - s_{2}) .

In order to obtain the innovation distribution, from here on, we assume that $s_{2} \geq s_{1} > 1$ . The rational function $\frac{U_{1} (s)}{(s - s_{1}) (s - s_{2})}$ in (13) can now be decomposed into partial fractions (see Feller, 2008) and the pgf of the innovation variable can be rewritten as

φ_{ϵ} (s)

=

1 + \frac{ρ_{1}}{s_{1} - s} + \frac{ρ_{2}}{s_{2} - s},

where

ρ_{1} = - \frac{U_{1} (s_{1})}{s_{1} - s_{2}} and ρ_{2} = - \frac{U_{1} (s_{2})}{s_{2} - s_{1}} .

Using the same geometric expansion for $\frac{1}{s_{1} - s}$ and $\frac{1}{s_{2} - s}$ as in Feller (2008), the pmf of the innovation sequence is obtained as

Pr [ϵ_{t} = m] = {\begin{matrix} π, & if & m = 0, (1 - π) [w_{1} (1 - p_{1}) p_{1}^{m - 1} + w_{2} (1 - p_{2}) p_{2}^{m - 1}], & if & m \geq 1, \end{matrix}

(14)

where $p_{1} = 1 / s_{1}$ , $p_{2} = 1 / s_{2} (p_{2} \leq p_{1})$ and $π = c / ¯ c$ . Note that $w_{1} (1 - p_{1}) p_{1}^{m - 1} + w_{2} (1 - p_{2}) p_{2}^{m - 1}$ is a mixture of two geometric distributions, where $w_{1} = p_{1} / (p_{1} - p_{2})$ and $w_{2} = p_{2} / (p_{2} - p_{1})$ are the weights with $w_{1} + w_{2} = 1$ . If $π \equiv p_{2} \equiv 0$ , then $ϵ_{t} \sim Geo (1 - p_{1})$ and if $p_{2} \equiv 0$ , then $ϵ_{t}$ has a hurdle geometric distribution with parameter $1 - p_{1}$ . Also, if $π \equiv p_{1} \equiv 0$ , then $ϵ_{t} \sim Geo (1 - p_{2})$ and if $p_{1} \equiv 0$ , $ϵ_{t}$ has a hurdle geometric distribution with parameter $1 - p_{2}$ .

Since in (14) the zeros come from the Bernoulli process with parameter $π$ 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

	$E [ϵ_{t}]$	$=$	$\frac{(1 - π)}{(1 - p_{1}) (1 - p_{2})},$
	$Var [ϵ_{t}]$	$=$	$\frac{(1 - π) [p_{1}^{2} p_{2} - 3 p_{1} p_{2} + p_{1} + p_{2} + π]}{(1 - p_{1})^{2} (1 - p_{2})^{2}},$

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, ${X_{t}}_{t \in Z}$ , with inflated-parameter geometric (Kolev et al. , 2000) marginals. The proposed process is based on the binomial thinning operator and satisfies the equation:

X_{t} = α \circ X_{t - 1} + ϵ_{t}, t \in Z,

with $α \in [0, 1)$ and ${X_{t}}_{t \in Z}$ being a stationary process with inflated-parameter geometric marginals, i.e., with pmf given by

Pr [X_{t} = m] = \frac{ρ}{μ + ρ} 1_{{0}} (m) + (1 - \frac{ρ}{μ + ρ}) {(\frac{μ + ρ}{1 + μ})}^{m} (\frac{1 - ρ}{1 + μ}), m = 0, 1, 2, \dots,

where $ρ \in [0, 1)$ and $μ > 0$ . The inflated-parameter geometric distribution is well-known as $ρ$ -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

φ_{X} (s) = \frac{1 - ρ s}{1 - s ρ + μ (1 - s)}, | s | < 1.

The mean and variance of the process ${X_{t}}_{t \in Z}$ are given, respectively, by

E [X_{t}] = \frac{μ}{1 - ρ} and Var [

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

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Regressions with Fractional d=1/2 and Weakly Nonstationary Processes

Simultaneous Transformation and Rounding (STAR) Models for Integer-Valued Data

1 Introduction

2 Main results

2.1 Method 1: Fractional decomposition of the innovation distribution

Remark 2.1.

Remark 2.2.

Remark 2.3.

Proof.

Example 1 (Geometric INAR(1) process).

Example 2 (New geometric INAR(1) process).

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

2.2 Method 2: Linear rational probability generation function

Example 4 (Two-parameter innovation sequence).

Example 5 (Zero-modified geometric innovation process).

2.3 Method 3: Quadratic rational probability generation function

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

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

Authors

Related Research

Stationary underdispersed INAR(1) models based on the backward approach

A series representation of the discrete fractional Laplace operator of arbitrary order

Integer-valued autoregressive process with flexible marginal and innovation distributions

A new integer valued AR(1) process with Poisson-Lindley innovations

A copula-based bivariate integer-valued autoregressive process with application

Regressions with Fractional d=1/2 and Weakly Nonstationary Processes

Simultaneous Transformation and Rounding (STAR) Models for Integer-Valued Data

This week in AI

1 Introduction

2 Main results

2.1 Method 1: Fractional decomposition of the innovation distribution

Remark 2.1.

Remark 2.2.

Remark 2.3.

Proof.

Example 1 (Geometric INAR(1) process).

Example 2 (New geometric INAR(1) process).

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

2.2 Method 2: Linear rational probability generation function

Example 4 (Two-parameter innovation sequence).

Example 5 (Zero-modified geometric innovation process).

2.3 Method 3: Quadratic rational probability generation function

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