Thinning-based models in the analysis of integer-valued time series: a review

Abstract

This article aims at providing a comprehensive survey of recent developments in the field of integer-valued time series modelling, paying particular attention to models obtained as discrete counterparts of conventional autoregressive moving average and bilinear models, and based on the concept of thinning. Such models have proven to be useful in the analysis of many real-world applications ranging from economy and finance to medicine. We review the literature of the most relevant thinning operators proposed in the analysis of univariate and multivariate integer-valued time series with either finite or infinite support. Finally, we also outline and discuss possible directions of future research.

Keywords

time series autocorrelation function(ACF)counts thinning operators

1 Introduction

Modelling and predicting the temporal dependence and evolution of time series of low counts have attracted a lot of attention over the last years. This is partially due to the increasing availability of relevant high-quality data sets in various fields of applications such as social science, industry, finance and economy, medicine and ecology just to mention a few. This great variety of application areas is illustrated by the data examples being plotted in Figures 1 and 2. Figure 1a shows a time series of monthly counts of sex offences (21st police car beat in Pittsburgh, 1990–2001) as discussed in Ristić et al. (2009). Figure 1b shows the monthly number of countries in the EA17 Euro area having stable prices (i.e., with an inflation rate below 2 %, period 2000–11); see Weiß and Kim (2014) for details.

Figure 2 shows examples of bivariate count data time series. The time series in Figure 2a-b (having the infinite range

ℕ_{0}^{2}

) are about the daily number of daytime and nighttime road accidents in Schiphol area (Netherlands) in the year 2001; see Pedeli and Karlis (2011) for a detailed description of the data. Figure 2c-d show the number of rainy days per week in the German towns of Bremen and Cuxhaven (period 2000–10), which constitutes a time series with the finite range

{0, \dots, 7}^{2}

; further details are provided by Scotto et al. (2014).

It is important to stress that there is no unifying approach applicable to modelling all the above time series of counts. Consequently, the analysis of such time series has to be restricted to special classes of integer-valued models. An useful conceptual division of these models can be made as being either observation-driven or parameter-driven models, though this distinction may be blurred Jung and Tremayne(2011a). A suitable class of observation-driven models is the one including models based on thinning operators. Models belonging to this class are obtained by replacing the multiplication in the conventional time series models (e.g., ARMA) by an appropriate thinning operator, along with considering a discrete distribution for the sequence of innovations in order to preserve the discreteness of the counts.

In modelling time series having an infinite range of counts, the class of INteger-valued AutoRegressive Moving Average (INARMA) models based on the binomial thinning operator of Steutel and van Harn (1979) plays a prominent role. Steutel and van Harn's binomial thinning operator (sometimes also referred to as ‘binomial subsampling’, see Puig and Valero 2007 takes the form

ϕ \circ X : = \sum_{i = 1}^{X} ξ_{i}, X > 0

(1.1)

and

0

otherwise, where

ξ : = (ξ_{i})

is a collection of independent and identically distributed (i.i.d.) Bernoulli counting random variables (r.v's) with (fixed) probability of success

ϕ \in [0; 1]

. Furthermore, it is assumed that

X

is a non-negative integer-valued random variable independent of

ξ

. Note that, conditioned on

X

ϕ \circ X

is binomially distributed, i.e.,

ϕ \circ X | X \sim Bi (X, ϕ)

. The binomial thinning operator 1.1 shares several properties with the multiplication operator, namely,

$ϕ_{1} \circ (ϕ_{2} \circ X) \overset{d}{=} (ϕ_{1} ϕ_{2}) \circ X \overset{d}{=} ϕ_{2} \circ (ϕ_{1} \circ X)$ ;

$E (ϕ_{1} \circ X) = ϕ_{1} E (X)$ ;

$ϕ_{1} \circ (Z + Y) \overset{d}{=} ϕ_{1} \circ Z + ϕ_{1} \circ Y$ , with the r.v's $Z, Y$ having support in $ℕ_{0}$ .

The last property above holds true provided that the counting sequences involved in

ϕ_{1} \circ Z : = \sum_{i = 1}^{Z} ξ_{i, Z}

and

ϕ_{1} \circ Y : = \sum_{i = 1}^{Y} ξ_{i, Y}

, are independent. Another important similarity between both operators, as quoted in Hall (2001), which is particularly useful for the analysis of the extremal properties of models involving the binomial thinning operator is the following: if

F_{X}

has a regularly varying right tail of the form

1 - F_{X} (n) = n^{- κ} L (n), κ \geq 0, n \in ℕ_{0},

for some slowly varying function

L (n)

+ \infty

, then the distribution tail of

ϕ \circ X

behaves asymptotically as the tail of

ϕ \cdot X

. Furthermore,

F_{X}

and

F_{ϕ \circ X}

are tail equivalent in the sense that

1 - F_{ϕ \circ X} (n) \sim ϕ^{κ} (1 - F_{X} (n))

, as

n \to \infty

. Unfortunately, such property does not hold for other important distributions namely those having exponential-type tails such as the geometric and the negative binomial distribution.

Turning now to the differences between the binomial thinning operator and the simple multiplication, the most important is that the distributive property does not hold for the binomial thinning operator since $ϕ_{1} \circ X + ϕ_{2} \circ X \overset{d}{\neq} (ϕ_{1} + ϕ_{2}) \circ X$ . Another important difference between both operators is due to the fact that

E [(ϕ \cdot X)^{2}] = ϕ^{2} \cdot E (X^{2}), but E [(ϕ \circ X)^{2}] = E [(ϕ \cdot X)^{2}] + ϕ (1 - ϕ) \cdot E (X),

which, in turn, implies that

V (ϕ \circ X) = V (ϕ \cdot X) + ϕ (1 - ϕ) \cdot E (X) .

Further properties of the binomial thinning operator can be found in Turkman et al. (2014), Weiß(2008a) and Silva and Oliveira (2004).

The INARMA model of order $p \geq 0$ and $q \geq 0$ based on the binomial thinning operator 1.1 is defined through the recursion

\begin{matrix} X_{t} - α_{1} \circ_{t} X_{t - 1} - \dots - α_{p} \circ_{t} X_{t - p} = Z_{t} + β_{1} \circ_{t} Z_{t - 1} + \dots + β_{q} \circ_{t} Z_{t - q}, t \in ℤ, \end{matrix}

(1.2)

where

(Z_{t})

is an i.i.d. sequence of integer-valued r.v's with finite mean and variance. Furthermore, it is assumed that all thinning operators are performed independently of each other and of

(Z_{t})

. The time index below the binomial thinning operator emphasizes the fact that the operators are performed at each time

t

. When

q = 0

in 1.2,

(X_{t})

is called an INAR of order

p

; when

p = 0

(X_{t})

is referred to as INMA of order

q

. In the subsequent sections, several subclasses of INARMA models will be explained in detail.

While INARMA models directly imitate the classical ARMA recursion, another approach useful to handle time series of counts is to consider generalized linear models, where the serial dependence structure is incorporated through a link function. This leads to a very flexible class of models, where, e.g., covariate information are easily incorporated. On the other hand, model identification and interpretation may become more problematic as well as closed-form formulae concerning the stationary marginal distribution. A commonly used regression model, with a linear link function, is the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) process of orders $p, q \in ℕ$ Heinen (2003); Ferland et al. (2006), defined as

X_{t} | F_{t - 1} : Po (λ_{t}); λ_{t} = α_{0} + \sum_{i = 1}^{p} α_{i} X_{t - i} + \sum_{j = 1}^{q} β_{j} λ_{t - j},

(1.3)

where

F_{t - 1} : = σ (X_{s}, s \leq t - 1)

α_{0} > 0

α_{i} \geq 0

and

β_{j} > 0

. The particular case

p = q = 1

was analyzed by Fokianos and Tjøstheim (2012) and Fokianos et al. (2009) under the designation of Poisson autoregression. The authors considered linear and non-linear representations for

λ_{t}

. We refer the reader to the surveys of Tjøstheim (2012) and Fokianos and Tjøstheim (2012) and the references therein for further details.

This article is intended to be an update of the survey published by Weiß(2008a) on the early developments in the analysis of univariate integer-valued time series based on thinning operators. It aims to provide a concise introduction to the subject with plentiful references. We review the thinning operators recently proposed in the literature (i.e., from 2008 onwards) as well as the thinning-based models commonly used in the analysis of univariate, and also multivariate integer-valued time series defined over bounded and unbounded sets. Efforts have been made to include all relevant contributions on the field to make this literature survey as complete and self-contained as possible, which has resulted in a rather voluminous list of references at the end of the paper.

The rest of the article is organized as follows: Section 2 surveys several classes of univariate binomial thinning-based models useful for representing integer-valued time series with finite and infinite support. Section 3 introduces several generalizations of the binomial thinning operator which are commonly used in the analysis of integer-valued time series exhibiting features, such as, non-stationarity, positive and negative autocorrelations, and overdispersion. Extensions of univariate thinning operators for the bivariate and multivariate case are discussed in Section 4, together with models based on such thinning operators to fitting bivariate/multivariate integer-valued time series defined over bounded and unbounded sets. Section 5 is devoted to conclusions and to discuss likely directions of future research.

2 Univariate binomial thinning-based models

A simple procedure to obtain models for integer-valued data is to replace the multiplication in conventional time series models, e.g., ARMA, bilinear or threshold models, by thinning operators in order to ensure the integer discreteness of the process. Among the most successful integer-valued time series models based on thinning operators are the INARMA $(p, q)$ models defined in 1.2. McKenzie (1985) and Al-Osh and Alzaid (1988) introduced the so-called stable INAR $(1)$ model,

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

(2.1)

with

α \equiv α_{1} \in [0; 1)

in 1.2, which constitutes an instance of a subcritical branching process with immigration. If the innovations

Z_{t}

have a finite mean, then the INAR

(1)

process in 2.1 is stationary Weiß (2013). The INAR

(1)

model shares many properties with the conventional AR

(1)

model, namely, the fact that for both models the autocorrelation function (ACF) takes the form

Corr (X_{t}, X_{t - k}) = α^{k}, k \in ℕ .

Note, however, that the range of variation of the ACF in the INAR

(1)

case is restricted to

[0; 1]

. Another important property of the INAR

(1)

model in 2.1 is that the discrete self-decomposable (DSD) distributions are possible marginal distributions, since the probability generating function (p.g.f.) of the INAR

(1)

model satisfies

P_{X} (s) = P_{X} (1 - α + α s) \cdot P_{Z} (s) .

(2.2)

Many well-known distributions belong to the class of DSD distributions, namely, the negative binomial, Poisson, generalized Poisson distribution, generalized discrete Mittag-Leffler distribution and discrete stable distributions as a sub-class. By appropriately varying the innovations’ distribution, also zero-modified or underdispersed marginal distributions can be obtained; see Weiß (2013), Jazi et al. (2012) and others. In view of the closeness property with regard to addition and binomial thinning as shown in Puig and Valero (2007), a very natural approach is to assume the innovations to be compound-Poisson-distributed Schweer and Weiß (2014). These options, however, do not include distributions defined on bounded sets, ruling out the possibility to consider, e.g. the binomial distribution as a marginal distribution for

X_{t}

. A modified AR(1)’like model for binomial counts making use of binomial thinning is discussed further in the text.

Empirically relevant generalizations of the stable INAR $(1)$ model have been suggested by several authors. For example, Silva et al. (2005) introduced the replicated INAR process being the $k$ th time series $X_{t, k}, (k = 1, \dots, r)$ , defined as

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

with

E [Z_{t, k}] < \infty

and

V [Z_{t, k}] < \infty

. In order to make the INAR models more flexible with respect to real data applications, it may be of interest to include also covariates in the model to account for the dependence of the thinning probabilities on several factors. For this purpose, Brännäs (1995) and Monteiro et al. (2008) considered the INAR

(1)

model in 2.1 and introduced the effect of explanatory variables through the thinning parameter

α

. Usually, the conventional specification

α_{t} = 1 / (1 + e^{y_{t} ω})

with

y_{t} = [y_{t, 1} \dots y_{t, l}]

and

ω = [ω_{1} \dots ω_{l}]^{'}

, is adopted. The explanatory variables vectors

y_{t}

are treated as fixed and

ω

represents the corresponding vectors of unknown parameters. Further important contributions are from Brännäs et al. (2002) (panel data) and Brännäs and Hellström (2001) (extended dependence structure). INAR

(1)

models contaminated with additive and innovational outliers were introduced and analyzed by Barczy et al. (2010); Barczy et al. (2012). Extensions of the INAR

(1)

model into the spatial context were considered by Ghodsi et al. (2012).

The counterpart to the conventional AR $(p)$ model with $p > 1$ , in the context of INARMA models, is the so-called INAR $(p)$ model which follows the recursion

X_{t} = \sum_{i = 1}^{p} α_{i} \circ_{t} X_{t - i} + Z_{t}, t \in ℤ .

(2.3)

We can distinguish three types of INAR

(p)

processes, namely, the stable case

(\sum_{i = 1}^{p} α_{i} < 1)

, the unstable case

(\sum_{i = 1}^{p} α_{i} = 1)

and the explosive case

(\sum_{i = 1}^{p} α_{i} > 1)

. We concentrate first on the stable case. Note that since the thinning operators are probabilistic, the joint distribution of

(α_{1} \circ_{t + 1} X_{t}, \dots, α_{p} \circ_{t + p} X_{t})

has to be considered leading to different types of INAR(

p

) models. Alzaid and Al-Osh (1990) assume a conditional multinomial distribution whereas Du and Li (1991) require conditional independence. It is important to note that the statistical properties of the Alzaid and Al-Osh (1990) model are very different from the properties of the model suggested by Du and Li (1991). In terms of second-order structure, for instance, the ACF resembles the one obtained for the usual ARMA

(p, p - 1)

model, whereas Du and Li's formulation implies that the ACF is the same as that of an AR(

p

) model. The stationarity condition for the INAR(

p

) process by Du and Li (1991) is that the roots of the characteristic polynomial

1 - α_{1} z - \dots - α_{p} z^{p} = 0

are outside the unit circle, which is equivalent to

\sum_{i = 1}^{p} α_{i} < 1

A major drawback of both representations for the INAR $(p)$ model is that its stationary marginal distribution, for $p \geq 2$ , differs from that for $p = 1$ . To tackle this problem, Zhu and Joe (2006) and Weiß(2008c) introduced the so-called Combined INAR( $p$ ) (CINAR( $p$ )) model, which is constructed by using a probabilistic mixing mechanism. If $(D_{t})$ is an i.i.d. process with $D_{t} = (D_{t, 1}, \dots, D_{t, p}) \sim MULT (1; ϕ_{1}, \dots, ϕ_{p})$ independent of $(Z_{t})$ , then the CINAR( $p$ ) process $(X_{t})$ is defined by the recursion

X_{t} = D_{t, 1} \cdot (α \circ_{t} X_{t - 1}) + \dots + D_{t, p} \cdot (α \circ_{t} X_{t - p}) + Z_{t}, t \in ℤ .

(2.4)

Assumptions concerning the joint distributions of the r.v's being involved in recursion 2.4 are discussed by Weiß(2008c). The main advantage of the CINAR(

p

) model when compared with the INAR(

p

) model is that if

(X_{t})

is a stationary CINAR(

p

) process then its p.g.f. satisfies 2.2. Hence, the possible marginal distributions of a stationary CINAR(

p

) process include the DSD family.

For the unstable case (i.e., when the characteristic polynomial has a unit root), Barczy et al. (2011) proved that under some natural assumptions, namely, the INAR $(p)$ process being primitive (i.e., $α_{p} > 0$ and the greatest common divisor of the set ${i \in (1, \dots, p) : α_{i} > 0}$ being equal to one) and that $E (Z_{1}^{2}) < \infty$ , the INAR $(p)$ process converges weakly towards a squared Bessel process (also known as Cox-Ingersoll-Ross process). The asymptotic behaviour of the unstable INAR $(p)$ process differs remarkably from that of the conventional unstable AR $(p)$ counterpart: the unit root of the characteristic polynomial has multiplicity one (contrary to the unstable AR $(p)$ model which may has real or complex unit roots with different multiplicities), and while the unstable INAR $(p)$ model converges to a well-defined limit, in contrast, however, for the unstable AR $(p)$ model such a limit does not exist in general. Further results for the unstable case can be found in Barczy et al. (2011); Barczy et al. (2014). Contrary to the other two cases, the explosive INAR $(p)$ process has been overlooked in the literature and remains as a topic for future investigation.

Discrete counterparts to the conventional $q$ th-order moving average processes (referred to as INMA( $q$ )) based on the binomial thinning operator have also been proposed in the literature. The INMA( $q$ ) model is defined by

X_{t} = \sum_{j = 0}^{q} β_{j} \circ_{t} Z_{t - j}, t \in ℤ,

(2.5)

where

(Z_{t})

is an i.i.d. sequence of non-negative integer-valued r.v's with finite mean and variance, and where

β_{0}, \dots, β_{q} \in [0; 1]

with

β_{q} \neq 0

and, usually,

β_{0} = 1

. In analogy to the INAR

(p)

case, each

Z_{t}

is involved in

q + 1

thinning operators, at

q + 1

different times

t, \dots, t + q

. Hence, for the same set of thinning parameters

(β_{0}, \dots, β_{q})

, a wide variety of models exhibiting very different autocovariance functions and joint distributions can be constructed by only changing the dependence structure of the thinning operators. Such types of INMA

(q)

models were proposed by Al-Osh and Alzaid (1988), McKenzie (1988) and Brännäs and Hall (2001). Weiß(2008b) showed that all these models can be embedded, indeed, into a single family of INMA(

q

) models.

Using the concept of binomial thinning, also conventional bilinear models can be adapted to the integer case leading to the class of integer-valued bilinear (INBL) models. This class of models is particularly suitable for modelling processes which assume low values with high probability, but exhibiting, at the same time, sudden bursts of large values. Drost et al. (2008) introduced the super diagonal INBL process, which is a particular class of the more general INBL $(p, q, m, n)$ class with $X_{t}$ given by the form

\begin{matrix} X_{t} = \sum_{i = 1}^{p} α_{i} \circ_{t} X_{t - i} + \sum_{j = 1}^{q} β_{j} \circ_{t} Z_{t - j} + \sum_{r = 1}^{m} \sum_{s = 1}^{n} γ_{r, s} \circ_{t} (X_{t - r} Z_{t - s}) + Z_{t}, t \in ℤ, \end{matrix}

(2.6)

where

γ_{r, s} = 0

r < s

and

(Z_{t})

is an i.i.d. sequence of non-negative, integer-valued r.v's having finite mean and variance. Doukhan et al. (2006) analyzed the special INBL

(1, 0, 1, 1)

model

X_{t} = α_{1} \circ X_{t - 1} + γ_{1, 1} \circ (X_{t - 1} Z_{t - 1}) + Z_{t}, t \in ℤ .

(2.7)

The stationary INBL

(1, 0, 1, 1)

process is capable of producing, despite its simplicity, sudden bursts of large values and hence is particularly suitable for modelling integer-valued time series showing heavy-tails of power law types (e.g., Zipf's law). Doukhan et al. (2006) and Drost et al. (2008) assume that the distinct binomial thinning operators involved in 2.6 and 2.7 are independent and also independent when applied to the different random variables. Figure 3 displays three simulated sample paths consisting in 1000 observations of the bilinear model 2.7 with

γ_{1, 1} = 0.2, 0.5, 0.7

and

α_{1} = 0.2

and

Z_{t} \sim

(1)

in all cases. As expected, when the bilinear parameter

γ_{1, 1}

tends to 1, bursts of large values become more obvious.

Drost et al. (2008) derived a condition guaranteeing the existence of a unique strictly stationary process satisfying 2.6 with finite first moment, and Doukhan et al. (2006) obtained conditions to ensure that the INBL

(1, 0, 1, 1)

process is second-order stationary. For the bilinear model in 2.7, Drost et al. (2008) also provided sufficient conditions for the existence of higher order moments of

X_{t}

. For these INBL processes, the ACF (when it exists) often provides little insight into the dependence structure of the process, and its empirical counterpart can behave in a very unpredictable way. Thus, second-order methods depending on the sample ACF for identification and estimation can misguide the practitioner and result in an inappropriate model being selected. It is worth also to mention here that conditions for invertibility of the general INBL

(p, q, m, n)

model (which are crucial for estimation and prediction) are not known. Therefore, the use of this class of models in practice is by far quite limited. Furthermore, the analysis of the probabilistic structure of INBL models (in particular, e.g., weak limits of extreme values) also remains an open question. This is partially due to the fact that it is difficult to obtain an explicit analytic expression for the stationary distribution of (2.6).

When the focus is on the analysis of integer-valued time series with a finite range of counts, say ${0, 1, \dots, n}$ , the models given above are useless. Addressing this issue, McKenzie (1985) gave a noticeable contribution by suggesting to replace the innovation term $Z_{t}$ in the INAR $(1)$ recursion 2.1 by the term $β \circ (n - X_{t - 1})$ , leading to the following representation for $X_{t}$ ,

X_{t} = α \circ X_{t - 1} + β \circ (n - X_{t - 1}), t \in ℤ,

(2.8)

with

α 1 pt = 1 pt β 1 pt + 1 pt ρ

β 1 pt = 1 pt π (1 1 pt - 1 pt ρ)

for

π \in (0; 1)

and

ρ \in (max {1 pt - 1 pt π / (1 - π), - (1 - π) / π}; 1),

where all thinning operators are performed independently of each other, and being the thinning operators at time

t

independent of

(X_{s})_{s < t}

. Note that the representation for

X_{t}

in 2.8 guarantees that the range of

X_{t}

is given by

{0, 1 \dots, n}

. The process in 2.8 used to be referred to as binomial AR

(1)

process and is a stationary Markov chain with

n + 1

states and binomial marginal distribution

Bi (n, π)

. Alternative correlated processes with binomial marginals based on the so-called hypergeometric thinning operator was introduced by Al-Osh and Alzaid (1991); see Weiß(2008a) for details. The binomial AR

(1)

process shares some properties with the conventional AR

(1)

models, namely, the fact that the ACF decays to zero at an exponential rate. Other important features of the binomial AR

(1)

process are that both the conditional mean and variance of

X_{t}

given

X_{t - 1}

are linear in

X_{t - 1}

, and the fact of being time-reversible. Closed-form expressions for the joint moments and cumulants were obtained by Weiß and Kim(2013a), whereas issues related with parameter estimation have been addressed by Cui and Lund (2010), Weiß and Pollett (2014) and Weiß and Kim(2013a); Weiß and Kim(2013b). Extensions for higher- order binomial models were first proposed by Weiß (2009). In order to preserve the range of

X_{t}

to be limited by

n

, the author introduced the class of combined binomial AR

(p)

processes in analogy with the class of CINAR

(p)

models in 2.4, by defining

(X_{t})

through the recursion

X_{t} = D_{t, 1} \cdot g_{1} (X_{t - 1}) + \dots + D_{t, p} \cdot g_{p} (X_{t - p}), t \in ℤ,

where

g_{i} (X_{t - i}) : = α \circ_{t} X_{t - i} + β \circ_{t} (n - X_{t - i})

for

i = 1, \dots, p

, with

α

and

β

defined as in 2.8, and

D_{t} : = (D_{t, 1}, \dots, D_{t, p}) \sim MULT (1; ϕ_{1}, \dots, ϕ_{p})

. All thinning operators at time

t

are performed independently of each other and of the i.i.d. process

(D_{t})

being

D_{t}

independent of all

X_{s}

and

g_{1} (X_{s}), \dots, g_{p} (X_{s})

, for

s < t

. A particularly relevant class of combined binomial AR

(p)

models is the one containing the so-called independent thinning models, because it leads to a conventional AR

(p)

autocorrelation structure while preserving the stationary marginal of the underlying first-order model.

3 Extensions of the binomial thinning operator

The binomial thinning operator 1.1 and the related time series models presented in Section 2 were generalized in a number of different ways. Such generalizations were motivated by the need for, e.g., capturing overdispersion, a modified probability for observing a zero, a particular type of dependence structure or including non-negative integers into the range of the process. In the present section, we concentrate on those generalizations leading to a univariate time series, while later in Section 4, we consider multivariate extensions.

3.1 Generalized thinning operators

Various authors have proposed modifications of the binomial thinning operator in 1.1 in order to make the integer-valued models based on thinning more flexible for practical purposes. Latour (1998) introduced the generalized thinning operator by allowing the components in $ξ$ to be i.i.d. integer-valued r.v's with finite mean and variance although not necessarily Bernoulli-distributed. Modifying the INAR(1) recursion 2.1 accordingly leads to the generalized INAR(1) (GINAR(1)) process, and Latour's GINAR $(p)$ model is the generalized counterpart of the INAR $(p)$ model by Du and Li (1991). The case in which $ξ_{i}$ ’s are i.i.d., r.v's with geometric distribution, say $G_{1}$ , was analyzed by Ristić et al. (2009) and referred to as the negative binomial thinning operator, say ‘ $⊙$ ’. The operator ‘ $⊙$ ’ is then used to modify the INAR(1) recursion 2.1. Ristić et al. (2009) showed that if the marginal distribution of $(X_{t})$ is also geometric (referred to as ‘new’ geometric INAR(1) process, NGINAR(1)), say $G_{2}$ , then $Z_{t}$ is a mixture of $G_{1}$ and $G_{2}$ . Nastić et al. (2012) proposed a $p$ th order autoregressive extension by adapting the CINAR( $p$ ) approach in equation 2.4. Ristić et al.(2012a) considered the case of negative binomial marginals, the NBINAR(1) process. A slightly modified type of negative binomial thinning (with an additional mixture) and related INARMA models are discussed by Aly and Bouzar(1994a); Aly and Bouzar(1994b).

Also the extended Poisson INAR $(1)$ model by Weiß (2015) can be understood as a special instance of Latour's model, but which uses binomial-Poisson thinning. In fact, also the (Poisson) INARCH $(1)$ model (i.e., the 1.3 with $p = 1$ and $q = 0$ ) may be subsumed under the class of AR $(1)$ -like thinning models (see, e.g., Grunwald et al. (1997); Weiß (2015) by considering the Poisson thinning operator. This argument also extends to the whole INARCH $(p)$ family, which can be seen as the corresponding instance of the GINAR $(p)$ model by Latour (1998). Another important special case of Latour's operator is the extended thinning operator proposed by Zhu and Joe (2003) being the $ξ_{j}$ ’s i.i.d. r.v's, independent of $X$ , having the same distribution as a random variable $Y$ with p.g.f.

E [s^{Y}] = \frac{(1 - ϕ) + (ϕ - γ) s}{(1 - ϕ γ) - (1 - ϕ) γ s}, γ \in (0; 1],

(3.1)

with mean

E [Y] = ϕ

and variance

V [Y] = ϕ (1 - ϕ) \frac{1 + γ}{1 - γ}

. Clearly, the binomial thinning operator corresponds to

γ = 0

in 3.1. Also the expectation thinning operator as proposed by Zhu and Joe (2010) is an instance of the generalized thinning operator, in which

ξ

constitutes a family of self-generalized r.v's with support on

ℕ_{0}

and

E [ξ_{j}] < 1

, for all

j = 1, \dots, X

Joe (1996) proposes a very general way of obtaining stationary INAR $(1)$ models of the form $X_{t} = M_{t} (X_{t - 1}, α) + Z_{t}$ , with $M_{t}$ being a random operator. The two terms on the right-hand side of the equation are assumed to be independent. Here, the idea is to properly select a marginal distribution and also an appropriate thinning operator, $M_{t}$ , to ensure that the marginal distribution of $X_{t}$ and $Z_{t}$ are from the same family. For this purpose, the author assumes that the marginal distribution of $X_{t}$ is an infinitely divisible exponential dispersion model with parameters $θ$ and $λ$ (in short ED $(θ, λ)$ ) which includes as special cases the Poisson, negative binomial and the generalized Poisson distribution. Furthermore, the random operator is such that $M_{t} (X_{t - 1}, α_{t}) | X_{t - 1} = x \sim G (α λ, (1 - α) λ, x),$ where $G$ is referred to as the contraction corresponding to ED. Joe (1996) proved that under this setting the marginal distribution of $M_{t}$ is also ED $(θ, α λ)$ and independent of $Z_{t} \sim ED (θ, (1 - α) λ)$ . Examples of contractions include the binomial, the quasi-binomial and the beta-binomial distributions. The quasi-binomial thinning of Alzaid and Al-Osh (1993) is a special case of the random operator of Joe (1996) in which the distributions of $X_{t}$ , $Z_{t}$ and $M_{t}$ are generalized Poisson with parameters $(θ, λ)$ , $(θ, (1 - α) λ)$ and $(θ, α λ)$ , respectively. Extensions of Joe's results for higher order INARMA models have been considered by Jørgensen and Song (1998) and Jung and Tremayne(2011b).

3.2 Random coefficient thinning

Joe (1996) and Zheng et al. (2006); Zheng et al. (2007) suggested to consider stochastic thinning operators by allowing $ϕ$ in 1.1 to be random itself. The resulting thinning operator is called random coefficient thinning. The concept of random coefficient thinning has been extended by Gomes and Canto e Castro (2009) by allowing a different discrete distribution associated to the thinning operator, in analogy to Latour's generalized thinning operator. Specifically, the authors define the generalized random coefficient thinning operator $ϕ \circ^{G} X$ , based on a discrete random variable $X$ , a random variable $ϕ$ with support on $ℝ_{+}$ , and a family of discrete-type distribution functions $G (μ, σ)$ parametrized by the mean $μ$ and the standard deviation $σ$ , as $ϕ \circ^{G} X | ϕ, X \sim G$ , with mean $μ = ϕ \cdot X$ and variance $σ^{2} = δ \cdot X$ , being $δ \equiv δ (ϕ, X)$ a function depending on $ϕ$ and $X$ . Possible choices for $G$ are: (i) Binomial $(X, ϕ) \Rightarrow δ = ϕ (1 - ϕ)$ ; (ii) Negative Binomial $(X, \frac{1}{1 + ϕ}) \Rightarrow δ = ϕ (1 + ϕ)$ ; (iii) Poisson $(ϕ X) \Rightarrow δ = ϕ$ ; and (iv) Geometric $(\frac{1}{ϕ X}) \Rightarrow δ = ϕ (ϕ X - 1)$ . Leonenko et al. (2007) generalized the concept of random coefficient thinning by introducing a class of mixed thinning operators, where the probability of success in 1.1 takes the form $ϕ = α A$ , with $α \in (0; 1)$ and $A$ being a random variable with distribution function concentrated in the interval $(0; 1)$ . The case $α \to 1$ , leads to the usual random coefficient thinning.

INAR $(1)$ models based on random coefficient thinning operators of the form

X_{t} = ϕ_{t} \circ X_{t - 1} + Z_{t}, t \in ℤ,

(3.2)

where

(ϕ_{t})

is a sequence of i.i.d. r.v's each one taking values in the interval

[0; 1)

, and

(Z_{t})

is a sequence of i.i.d. integer-valued non-negative r.v's, independent of

(ϕ_{t})

, were considered by Zhang and Wang (2015), Roitershtein and Zhong (2013), Bakouch and Ristić (2010), Gomes and Canto e Castro (2009) and Zheng et al. (2007). Gomes and Canto e Castro (2009) and Zheng et al. (2007) proved that if

E (Z_{t})

and

V (Z_{t})

are finite, then the condition

E (ϕ_{t}^{2}) < 1, \forall t \in ℤ

, ensures that there exists a unique non-negative integer-valued weakly stationary process satisfying 3.2. Leonenko et al. (2007) analyzed the INAR

(1)

model based on the mixed thinning operator, i.e., by replacing the operator

α \circ

by the mixed operator

α A_{t} \circ

in 2.1, where

(A_{t})

is a sequence of i.i.d. r.v's concentrated on the interval

(0; 1)

with common distribution

F_{A}

. The authors proved that if process has finite first and second moments then

Corr (X_{t}, X_{t - k}) = (α E [A])^{k}, k \in ℕ .

Moreover, extensions for the INMA model in 2.5 based on stochastic thinning sequences were considered by Hall et al. (2010), who assume that the

β_{j}

’s form a sequence of independent r.v's independent of

(Z_{t})

. Hall et al. (2010) analyzed the extremal properties (in particular the limiting distribution of the normalized maxima) of this class of INMA models.

For modelling of count data time series with a finite range, Weiß and Kim (2014) introduced a beta-binomial autoregressive model using the concept of random coefficient thinning. This particular model generalizes the binomial AR(1) model 2.8 to the case of finite counts showing extra-binomial variation (overdispersion with respect to the binomial distribution). The definition of the beta-binomial AR(1) model resembles the one in 2.8 by replacing the thinning operators $α \circ$ and $β \circ$ by $α_{ϕ} \circ$ and $β_{ϕ} \circ$ which are assumed to be r.v.’s following a Beta distribution with parameters $(α ϕ^{'}, (1 - α) ϕ^{'})$ and $(β ϕ^{'}, (1 - β) ϕ^{'})$ being $ϕ^{'} : = (1 - ϕ) / ϕ$ , respectively.

3.3 Thinning operators with dependence structure

All thinning operators above rely upon the assumption of independence across the counting variables. Extensions of binomial thinning based on Bernoulli-distributed dependent r.v's were proposed by Brännäs and Hellström (2001), and more recently by Ristić et al. (2013) who assume that the variables in 1.1 take the form

ξ_{i} = (1 - V_{i}) W_{i} + V_{i} Y, i \in ℕ,

(3.3)

where

(W_{i})

and

(V_{i})

are sequences of i.i.d. Bernoulli r.v's with parameters

ϕ \in [0; 1]

and

θ \in [0; 1]

, respectively, and

Y

is also a Bernoulli random variable with parameter

ϕ \in [0; 1]

. Furthermore, it is assumed that

W_{j}

V_{i}

and

Y

are independent for all

i, j \in ℕ

. The representation in 3.3 implies that

(ξ_{j})

forms a sequence of dependent Bernoulli r.v's with parameter

ϕ \in [0; 1]

, since

Corr (ξ_{i}, ξ_{j}) = θ^{2} \neq 0

for

θ \neq 0

and

i \neq j

. The case

θ = 0

corresponds to the binomial thinning operator. Notice that equation 3.3 also relates to the Pegram's operator Puig and Valero (2007) upon which Jacobs and Lewis (1983) and Biswas and Song (2009) described a unified framework to yield discrete-valued ARMA models. Pegram's operator is defined as follows: given two independent discrete r.v's

U

and

V

, and a mixing weight

ϕ \in (0, 1)

, Pegram's operator mix

U

and

V

with

ϕ

and

1 - ϕ

, to produce a new random variable

Z = (U, ϕ) * (V, 1 - ϕ)

, with probability mass function

P (Z = n) = ϕ P (U_{n} = n) + (1 - ϕ) P (V = n), n \in ℕ_{0} .

Another type of additional dependence structure is obtained through the assumption of time-dependence of the thinning parameters. Monteiro et al. (2010) considered an INAR

(1)

model based on the binomial thinning operator with periodically varying parameter of the form

X_{t} = a_{t} \circ X_{t - 1} + Z_{t},

with

a_{t} = α_{1}^{(j)} \in [0; 1)

for

t = j + kT, (j = 1, \dots, T, k \in ℕ_{0})

. Such models are useful, e.g. in the analysis of fire activity. On the other hand, Brännäs and Nordström (2006) proposed the following modification of the binomial AR

(1)

process in 2.8 to modelling daily accommodation time series for hotels and cottages,

X_{t} = α \circ X_{t - 1} + β \circ (n_{t} - α \circ X_{t - 1}), t \in ℤ .

(3.4)

Note that the parameter

n

is allowed to vary in time. The authors assume that

X_{t}

in 3.4 represents the number of occupied rooms in some hotel at day

t

, and the parameter

n_{t}

represents the hotel capacity at day

t

. Brännäs and Nordström (2006) also consider the situation in which the thinning parameters

α

and

β

in 3.4 are time-dependent variables, aiming to capture seasonal patterns in guest night time series as well as the income level of the country of origin and exchange rates. Some authors suggest to generalize the binomial thinning mechanism by allowing the thinning probabilities to dependent on previous observations (i.e., state-dependent thinnings). One such example is the functional coefficient INAR(1) (FINAR(1)) model as proposed by Triebsch (2008). She assumed the thinning parameter at time

t

in recursion 2.1 to be a measurable (and typically decreasing) function of

X_{t - 1}

, i.e.,

α_{t} = α (X_{t - 1})

, and established conditions for the weak dependence of the resulting FINAR

(1)

process. In the discussed data applications, a logistic form is assumed for

α (x)

. A related approach for the case of finite counts was considered by Weiß and Pollett (2014) who introduced the density-dependent binomial AR

(1)

process

X_{t} = α_{t} \circ X_{t - 1} + β_{t} \circ (n - X_{t - 1}), t \in ℤ .

Unlike the binomial AR

(1)

model in 2.8, the thinning parameters

α

and

β

are now presumed to be time-dependent, varying according to the ‘density’

X_{t} / n

. The authors derived a number of results concerning large approximations, law of large numbers and central limit theorems. Furthermore, the authors illustrate the usefulness of their approach by considering two particular subfamilies of density-dependent binomial AR

(1)

models which permit a wide range of overdispersion and underdispersion scenarios as well as positive and negative autocorrelations.

Another approach of state-dependence is given through the concept of self-exciting threshold models being motivated by the so-called piecewise phenomenon. The fundamental reason for introducing such classes of models is the need to model random cyclic behaviour that exists in many time series. In the continuous-valued case, threshold models are typically characterized by having a linear (ARMA) structure in each regime. The basic idea is as follows: we start with a linear model for $X_{t}$ and allow the parameters to vary according to the values of a finite number of past values of $X_{t}$ (self-exciting), or a finite number of past values of an associate process. Basic results on the probabilistic structure of this class of models can be found in Turkman et al. (2014) and Tong (1990). For dealing with time series of counts exhibiting piecewise-type patterns, Monteiro et al. (2012) introduced a class of self-exciting threshold INAR (SETINAR) models of order one and two regimes based on the binomial thinning operator. It falls within the class of the two-regimes SETINAR model of orders $p^{(1)}, p^{(2)} \in ℕ$ (SETINAR $(2, p^{(1)}, p^{(2)}))$ defined as

X_{t} = \{\begin{matrix} \sum_{i = 1}^{p^{(1)}} α_{i}^{(1)} \circ X_{t - i} + Z_{t}^{(1)}, X_{t - d} \leq R \\ \sum_{i = 1}^{p^{(2)}} α_{i}^{(2)} \circ X_{t - i} + Z_{t}^{(2)}, X_{t - d} > R \end{matrix},

(3.5)

where it is assumed that

\sum_{i = 1}^{p^{(j)}} α_{i}^{(j)} \in (0; 1)

, for

j = 1, 2

, and that the i.i.d. innovation sequences

(Z_{t}^{(1)})

and

(Z_{t}^{(2)})

have distributions

F_{1}

and

F_{2}

ℕ_{0}

, respectively. The constant

R

represents the threshold level of the process and the regime switch is triggered by the lag-

d

value of the series. Furthermore, the binomial thinning operators take the form

α_{i}^{(j)} \circ X_{t - i} : = \sum_{k = 1}^{X_{t - i}} ξ_{k, t, j}, j = 1, 2, i = 1, \dots, p^{(j)},

where

(ξ_{k, t, j})

, for

j = 1, 2

, are sequences of i.i.d. Bernoulli r.v's, independent of

(Z_{t}^{(1)})

and

(Z_{t}^{(2)})

, with success probabilities

α_{i}^{(1)}

and

α_{i}^{(2)}

, respectively. Monteiro et al. (2012) proved the existence of a strictly stationary process satisfying 3.5 for the case

p^{(1)} = p^{(2)} = 1

3.4 Thinning operators for

ℤ

-valued time series

All approaches discussed so far consider the case of count data time series, i.e., for time series having the non-negative integers as their range. Arguably, the first paper considering times series with possible values in the full set of integers, from now on abbreviated as $ℤ$ -valued time series, is the one by Kim and Park (2008). These authors introduced the signed binomial thinning operator being defined as follows:

ϕ \oplus X : = sgn (ϕ) \cdot sgn (X) \cdot \sum_{i = 1}^{| X |} ξ_{i}, | ϕ | < 1,

(3.6)

where

sgn (x) = 1

x \geq 0

and

- 1

otherwise, and

(ξ_{i})

is defined as in 1.1 but having probability of success

| ϕ |

. Based on their thinning operator, Kim and Park (2008) extended the INAR

(p)

model in 2.3 to fit integer-valued time series taking values on

ℤ

. Bakouch and Ristić (2010) used this operator to define a first-order model with Skellam-distributed innovations. The signed binomial thinning operator of Kim and Park (2008) has been generalized in many ways. Kachour and Truquet (2011), e.g., define the generalized signed thinning operator as

F * X : = sgn (X) \cdot \sum_{i = 1}^{| X |} ξ_{i},

(3.7)

X \neq 0

and 0 otherwise. The

ξ_{i}

’s have a common distribution

F

on a subset of

ℤ

and are independent of

X

, in analogy to the generalized thinnings being discussed in Section 3.1. Kachour and Truquet (2011) used the signed-type thinning operator 3.7 to extend the INAR

(p)

model in 2.3 to fit

ℤ

-valued time series. In particular, they introduced the signed INAR(1) (SINAR

(1)

) process by the recursion

X_{t} = F * X_{t - 1} + Z_{t},

with

(Z_{t})

taking values on

ℤ

and

F

charging zero with positive probability and

| E [ξ_{1}] | < 1

. Chesneau and Kachour (2012) analyzed the particular SINAR

(1)

model in which the distribution

F

is defined as follows:

P (ξ_{1} = - 1) = (1 - ϕ)^{2}

P (ξ_{1} = 0) = 2 ϕ (1 - ϕ)

and

P (ξ_{1} = 1) = ϕ^{2}

, where

ϕ \in (0; 1)

. Note that

ξ_{1} \overset{d}{=} υ_{1} - 1

with

υ_{1} \sim

(2, ϕ)

and

P (\sum_{j = 1}^{k} ξ_{j} = r) = P (υ_{k} = k + r), r \in {- k, \dots, k}, υ_{k} \sim Bi (2 k, ϕ) .

Alzaid and Omair (2014) introduced the extended binomial thinning operator defined as

S_{ϕ, γ} (X) : = sgn (X) \cdot \sum_{i = 1}^{| X |} ξ_{i} + \sum_{i = 1}^{Y} η_{i}, ϕ \in [0; 1],

(3.8)

where

X

is a

ℤ

-valued random variable and

Y | X = x

is Bessel-distributed with parameters

| x |

and

γ

. Moreover,

(ξ_{i})

is defined as in 1.1 being independent of

(η_{i})

X

and

Y

. Furthermore,

(η_{i})

is a sequence of i.i.d. r.v's independent of

X

and

Y

, and

η_{i}

has the probability mass distribution

P (η_{i} = 1) = P (η_{i} = - 1) = ϕ (1 - ϕ)

and

P (η_{i} = 0) = 1 - 2 ϕ (1 - ϕ)

. Alzaid and Omair (2014) proved that within this framework,

S_{ϕ, γ} (X) | X = x

follows an extended binomial distribution Alzaid and Omair (2012) with parameters

x, ϕ

and

γ

. Note that the extended binomial thinning operator includes the binomial operator in 1.1 provided that

X

is non-negative and

γ = 0

in 3.8. Operator 3.8 is then used by Alzaid and Omair (2014) to define the Poisson difference INAR(1) process by

X_{t} = δ S_{ϕ, γ} (X_{t - 1}) + Z_{t}, t \in ℤ,

where

(Z_{t})

is an i.i.d. sequence of r.v's with Skellam distribution and

δ

is a parameter taking values

1

and

- 1

describing the sign of the correlation. Alzaid and Omair (2014) proved that under the above conditions, the marginal distribution of

X_{t}

is also Skellam.

Kim and Park's operator was also extended by Zhang et al. (2010) who introduced the signed generalized power series thinning operator. Zhang and co-workers’ operator is defined as in 3.6 but with the $ξ_{i}$ ’s being i.i.d. with a generalized power series distribution. Such family of distributions includes, among others, the Poisson, the logarithmic series, the binomial and the negative binomial one.

An important advantage of the aforementioned models is that their ACF preserves the structure of the ACF of conventional ARMA models which, in turn, implies that their ACF will alternate between positive and negative values provided that some of models’ parameters are negative (in particular, $ϕ$ in 3.6, the mean of the distribution $F$ in 3.7, and parameter $δ$ in Alzaid and Omair's model). Hence, such models are good candidates for fitting $ℤ$ -valued time series with some negative autocorrelations.

Another approach for obtaining an operator acting on a truly integer-valued random variable was proposed by Freeland (2010). Freeland's operator is defined as follows: let $Z$ be the difference of two latent and independent Poisson random variables $X$ and $Y$ with the same mean $μ$ , then $Z = X - Y$ follows the Skellam distribution with parameter $μ$ . The corresponding thinning operator ‘ $★$ ’ is defined as $α ★ Z | Z : = α \circ X - α \circ Y | X - Y,$ for $α \in [0; 1)$ , which is then used to define the so-called True INAR(1) model, abbreviated as TINAR $(1)$ . The TINAR $(1)$ model can be expressed as the difference of two independent Poisson INAR $(1)$ models. This concept was also adapted to the negative binomial thinning operator Ristić et al. (2009) by Barreto-Souza and Bourguignon (2015) and Nastić et al. (2014). The corresponding first-order model, the skew TINAR $(1)$ model, has observations following a skew discrete Laplace distribution.

Before concluding this section a reference should be made to the two random operators (referred to as first- and second-order random rounding operators) introduced by Liu and Yuan (2013). The usefulness of such operators rely mainly on the fact that the conditional mean and the conditional variance of the process are modelled separately, which implies that no assumptions on the relationship between the conditional mean and variance are needed. This is in contrast, e.g., to the class of binomial thinning-based INARMA models and the class of INGARCH models in which strong assumptions between the conditional mean and variance are imposed. Further advantages of models based on Liu and Yuan's operators include their flexibility in handling $ℤ$ -valued time series with range of variation of the ACF in $[- 1; 1]$ .

3.5 Parameter estimation, testing and forecasting

A large number of articles cover aspects of parameter estimation for INAR-type models. For first- order models, also results from the literature about branching processes with immigration can often be adopted (see Weiß (2015) for some references). But also for higher- order models, types of, e.g., moment estimators, conditional least-squares (CLS) estimators or maximum likelihood (ML) estimators have been developed, and their asymptotic properties have been discussed, see, e.g., Latour (1998) (CLS estimation) or Bu et al. (2008) (ML estimation) for the case of GINAR $(p)$ models. It is well known, however, that the CLS method is statistically inefficient and that the ML method is difficult to implement, mainly due to the complication in computing the transition probabilities involved in the log-likelihood function, as the order of the model increases. To overcome such difficulties, Pedeli et al. (2014) proposed a saddlepoint-based technique to obtain accurate approximation of the transition probabilities. A semiparametric estimation approach for INAR $(p)$ models was proposed and investigated by Drost et al. (2009), while Weiß and Kim(2013b) consider diverse estimation approaches for binomial AR(1) models. For the INBL models, in contrast, only the method of moments has been investigated so far as an estimation technique.

Also model identification and diagnostic techniques have been proposed in the literature for INARMA models, while corresponding approaches for INBL models are currently not available. Jung and Tremayne (2003), for instance, compared several tests for uncovering significant serial dependence as being caused by INARMA processes. Sun and McCabe (2013) developed further such tests based on the likelihood score for diverse types of innovations’ distribution, while Schweer and Weiß (2014) proposed a dispersion test for INAR(1) models. Hudecová et al. (2015) discuss tests based on the empirical p.g.f. Concerning time series of counts with a finite range, we find, e.g., a test for uncovering extra-binomial variation in binomial AR(1) processes in the work by Weiß and Kim (2014), while Kim and Weiß (2015) investigate a number of goodness-of-fit tests for binomial AR(1) processes.

Procedures for constructing forecasts for INAR-type models have been also proposed in the literature. The traditional way of producing forecasts is via the $h$ -step-ahead conditional distribution of $X_{n + h}$ given $X_{n}$ , $p_{h} (x | X_{n}) : = P (X_{n + h} = x | X_{n})$ . Forecasting can be pursued within the setting of conditional expectations in which case the $h$ -step-ahead predictor yielding minimum mean squared error forecasts is ${\hat{X}}_{n + h} = E (X_{n + h} | X_{n})$ . The major drawback of forecasting based on the conditional expectation is that it hardly produces coherent (i.e., integer-valued) predictions. In order to generate data coherent predictions, the median of the $h$ -step-ahead conditional distribution (which minimizes the expected absolute error) or its corresponding mode can be employed as a point forecast. However, as pointed out by Freeland and McCabe (2004), using the median or the mode may be potentially misleading specially when the range of variation of $X_{n + h}$ is small. In such cases, a more enlightened approach is to provide point estimates and confidence intervals for $p_{h} (x | X_{n})$ , i.e., the probability of occurrence of $x$ , instead of obtaining forecasts based on the conditional mean or median. More recently, McCabe et al. (2011) proposed a novel method for producing efficient probabilistic forecasts in the INAR $(p)$ class in 2.3. The method involves estimating the forecast distribution of the INAR $(p)$ model for a given horizon and to quantify the sampling variation that is associated with future counts, via subsampling techniques. McCabe and co-authors’ technique for assessing the effect of sampling variation mimics the conventional prediction intervals, but ensures at the same time, that the non-negativity and the unit property of the forecast distribution still stands true. Furthermore, the authors also proved that, under certain conditions, the estimated forecast distribution based on the non-parametric ML estimates for the model's parameters $α_{1}, \dots, α_{p}$ and $F_{Z}$ , is an asymptotic efficient estimator of the forecast distribution in the sense of the Hajek convolution theorem.

4 Multivariate models based on thinning operators

4.1 Multinomial thinning

Extensions of univariate thinning operators for the multivariate case is nowadays a topic of active research and of great applicative interest. One of the first approaches towards a multivariate thinning mechanism is the one by McKenzie (1988). In a first step, binomial thinning is generalized to $α * X : = \sum_{j = 1}^{X} ξ_{j},$ where $α : = (α_{1}, \dots, α_{d})^{'} \in (0; 1)^{d}$ is a vector of probabilities with $\sum_{j = 1}^{d} α_{j} < 1$ . The $ξ_{j}$ are i.i.d. random vectors being independent of $X$ , which are multinomially distributed according to $MULT (1; α_{1}, \dots, α_{d})$ . Obviously, $d = 1$ corresponds to the binomial thinning operator in 1.1. In the second step, a matrix $A \in (0; 1)^{d \times d}$ with column vectors $α_{i} : = (α_{i 1}, \dots, α_{id})^{'}$ as above is defined. If the random vector $X$ has range contained in $ℕ_{0}^{d}$ , then

A * X : = \sum_{i = 1}^{d} α_{i} * X_{i} \equiv \sum_{i = 1}^{d} \sum_{j = 1}^{X_{i}} ξ_{ij},

(4.1)

is also a random vector having a range contained in

ℕ_{0}^{d}

. It is assumed that all thinning operators are mutually independent and independent of

X

. Under these assumptions, the thinning operator 4.1 is referred to as multinomial thinning. Like for binomial thinning, for the multinomial thinning it also holds that

α * (X + Y) \overset{d}{=} α * X + α * Y

and

A * (X + Y) \overset{d}{=} A * X + A * Y .

Multinomial thinning operators were used by McKenzie (1988) to construct types of multivariate Poisson ARMA models. McKenzie's multinomial thinning approach was extended by Aly and Bouzar(1994b) by using a type of negative binomial thinning, also see Section 3.1.

4.2 Matrix-binomial thinning

Franke and Subba Rao (1993) introduced a multivariate INAR(1) model based upon independent binomial thinning operators. An extension for multivariate integer-valued autoregressive models based on generalized thinning operators (see Section 3.1) was proposed by Latour (1997). The thinning concept of Franke and Subba Rao (1993) is as follows: let $X : = [X_{1} X_{2} \dots X_{d}]^{'}$ be a random vector with values in $ℕ_{0}^{d}$ . For a given $(d \times d)$ -matrix $Φ : = [ϕ_{i, j}]$ with $ϕ_{i, j} \in [0; 1]$ , the matrix thinning $Φ \circ X$ gives a $d$ -dimensional random vector with $i$ th component

[Φ \circ X]_{i} : = \sum_{j = 1}^{d} ϕ_{i, j} \circ X_{j} \equiv \sum_{j = 1}^{d} \sum_{r = 1}^{X_{j}} ξ_{i, j, r}, i = 1, \dots, d,

where

(ξ_{i, j, r})

is a sequence of i.i.d. Bernoulli r.v's with probability of success

ϕ_{i, j} \in [0; 1]

. Furthermore, it is assumed that the thinning operators are performed independently of each other. Karlis and Pedeli (2013) and Pedeli and Karlis (2011); Pedeli and Karlis(2013a); Pedeli and Karlis(2013b) restrict to the diagonal case

ϕ_{21} = ϕ_{12} = 0

such that

[Φ \circ X]_{i} \overset{d}{=} ϕ_{ii} \circ X_{i}

for

i = 1, 2

, i.e., the marginals behave like the univariate binomial thinning operator in 1.1. However, this nice feature is obtained at the cost of no additional cross-correlation between

[Φ \circ X]_{1}

and

[Φ \circ X]_{2}

. On the contrary, if

ϕ_{12}, ϕ_{21} \neq 0

is allowed as in Franke and Subba Rao (1993) and Boudreault and Charpentier (2011), then the marginals of

Φ \circ X

no longer behave like the univariate binomial thinning operators. Pedeli and Karlis (2011) introduced the bivariate INAR(1) (BINAR

(1)

) model with bivariate Poisson and bivariate negative binomial innovations. Their model is defined as

X_{t} = A \circ X_{t - 1} + Z_{t} \equiv [\begin{matrix} ϕ_{1} & 0 \\ 0 & ϕ_{2} \end{matrix}] \circ [\begin{matrix} X_{t - 1, 1} \\ X_{t - 1, 2} \end{matrix}] + [\begin{matrix} Z_{t, 1} \\ Z_{t, 2} \end{matrix}], t \in ℤ,

(4.2)

where

[Z_{t, 1} Z_{t, 2}]^{'}

are assumed to be independent

ℕ_{0}^{2}

-valued random pairs. In order to also account for negative correlation between the time series, Karlis and Pedeli (2013) suggest to use an appropriate bivariate copula for the innovations. Note that, one of the drawbacks of the model 4.2 is that the autoregression matrix

A

is diagonal, which means that the thinning operation causes no cross-correlation in the counts. The more general

d

-variate INAR

(1)

process based upon the assumption that

Φ

is a non-diagonal matrix was introduced by Franke and Subba Rao (1993) and later by Boudreault and Charpentier (2011) and Pedeli and Karlis(2013c), as

X_{t} = A \circ X_{t - 1} + Z_{t}, t \in ℤ,

(4.3)

for some

(d \times d)

-matrix

A

with entries in

[0; 1]

, and some i.i.d. random vector

Z_{t}

taking values in

ℕ_{0}^{d}

. Franke and Subba Rao (1993) derived criteria ensuring the existence of a strictly stationary

d

-variate INAR(1) process satisfying 4.3. Quoreshi (2006) applied the non-diagonal matrix-binomial thinning operators to define a multivariate version of the INMA

(q)

model according to 2.5; stochastic properties of this model are derived for the bivariate case.

While matrix-binomial thinning was successfully applied to the case of count data time series with an infinite range, one observes difficulties if trying to extend the binomial AR $(1)$ model in 2.8 for the bivariate case. It is quite tempting to consider the following representation for $X_{t}$ ,

X_{t} = A \circ X_{t - 1} + B \circ (n - X_{t - 1}),

(4.4)

where

n : = [n_{1} n_{2}]^{'}

. If the matrices

A

and

B

are diagonal ones (as in Pedeli and Karlis’ models), then no cross-correlation will be observed because no innovations are included in model recursion 4.4. On the other hand, if the matrices

A

and

B

are non-diagonal, then

[A \circ X_{t - 1}]_{i} \leq X_{t - 1, i}

and

[B \circ (n - X_{t - 1})]_{i} \leq n_{i} - X_{t - 1, i}

, for

i = 1, 2

, do not necessarily hold anymore, so it may happen that

A \circ X_{t - 1} + B \circ (n - X_{t - 1}) \notin {0, \dots, n_{1}} \times {0, \dots, n_{2}}

. Such kind of problems are avoided if the concept of bivariate binomial thinning is used.

4.3 Bivariate binomial thinning

A different generalization of the binomial thinning operator 1.1 for the bivariate case, based on the bivariate binomial distribution of type II (BVB $_{II}$ ) Kocherlakota and Kocherlakota (1992), has been recently introduced by Scotto et al. (2014). This new operator has a number of advantages compared to the approach of the previous Section 4.2, see the details below. Scotto et al. (2014) define the bivariate binomial thinning operator as

α \otimes X ∣ X \sim {BVB}_{II} (X_{1}, X_{2}, min {X_{1}, X_{2}}; α_{1}, α_{2}, ϕ_{α}),

(4.5)

where

α : = (α_{1}, α_{2}, ϕ_{α})

with

0 < α_{1}, α_{2} < 1

and

ϕ_{α}

defined on a proper set (see Scotto et al. (2014) p. 235). Note that for this operator, it follows that

[α \otimes X]_{1} \sim Bi (X_{1}, α_{1})

and

[α \otimes X]_{2} \sim Bi (X_{2}, α_{2})

, i.e., marginally, it behaves like the univariate binomial thinning operator. Furthermore, the bivariate binomial thinning operator induces cross-correlation as long as

ϕ_{α} \neq 0

, and the additional cross-correlation may be both positive or negative provided that

ϕ_{α} > 0

ϕ_{α} < 0

, respectively. Note that the diagonal matrix thinning used by Karlis and Pedeli (2013) and Pedeli and Karlis (2011); Pedeli and Karlis(2013a); Pedeli and Karlis(2013b) is a special case of 4.5 with

ϕ_{α} = 0

Bivariate binomial thinning can also be utilized for the case of finite counts. To tackle this task, Scotto et al. (2014) introduced the bivariate binomial AR(1) model (BVB $_{II}$ -AR(1)), where the bivariate process $(X_{t})$ with $X_{t} : = [X_{t, 1} X_{t, 2}]^{'}$ satisfies the following recursion $X_{t} = α \otimes X_{t - 1} + β \otimes (n - X_{t - 1}),$ where the thinning operators are performed independently of each other. Here, $α : = (α_{1}, α_{2}, ϕ_{α})$ , and $β : = (β_{1}, β_{2}, ϕ_{β})$ are the thinning parameters defined as $β_{i} : = π_{i} \cdot (1 - ρ_{i})$ and $α_{i} : = β_{i} + ρ_{i}$ , where $π_{1}, π_{2} \in (0; 1)$ and $ρ_{i} \in (max {- \frac{π_{i}}{1 - π_{i}}, - \frac{1 - π_{i}}{π_{i}}} 1)$ for $i = 1, 2$ . It is important to stress that the marginals of the BVB $_{II}$ -AR $(1)$ process are distributed like the usual binomial AR $(1)$ process according to 2.8.

4.4 A bivariate extension of negative binomial thinning

A bivariate extension of the random coefficient thinning approach (see Section 3.2) was first considered by Ristić et al.(2012b). Their approach is based on the negative binomial thinning operator ‘ $⊙$ ’ discussed in Section 3.1, defined as

\begin{matrix} U ⊙ X = [\begin{matrix} U_{1} ⊙ X_{1} + U_{2} ⊙ X_{2} & V_{1} ⊙ X_{1} + V_{2} ⊙ X_{2} \end{matrix}]^{'}, \end{matrix}

(4.6)

where

(U_{1}, U_{2})

and

(V_{1}, V_{2})

are mutually i.i.d. random vectors with

P (U_{1} = ϕ_{11}, U_{2} = 0) = 1 - P (U_{1} = 0, U_{2} = ϕ_{11}) = p

and

P (V_{1} = ϕ_{22}, V_{2} = 0) = 1 - P (V_{1} = 0, V_{2} = ϕ_{22}) = q

, with

p, q \in [0; 1]

. Hence, the operator 4.6 can also be expressed as a mixture of negative binomial thinnings. Ristić et al.(2012b) introduced a bivariate INAR(1) model for

(X_{t})

with positively correlated geometric marginals based on the thinning operator 4.6. This model can also be understood as some kind of mixture of two NGINAR(1) processes Ristić et al. (2009).

4.5 Signed matrix thinning

Bulla et al. (2012) introduced the so-called signed matrix (SM) thinning operator as an extension of the signed thinning operator in 3.7 for the bivariate case. The SM thinning operator is defined as $F * X = [F_{11} * X_{1} + F_{12} * X_{2} F_{21} * X_{1} + F_{22} * X_{2}]^{'}$ , where $F_{ij}$ represents the common distribution of $ξ_{i, j, •} : = {[ξ_{i, j, 1} \dots ξ_{i, j, X_{j}}]}^{'}$ , for $(i, j) \in {(1, 2)}^{2}$ . Furthermore, it is assumed that all counting sequences associated with ‘ $F_{ij} *$ ’, $ξ_{i, j, •}$ , are mutually independent.

Bulla et al. (2012) introduced the class of bivariate signed INAR(1) (B-SINAR $(1)$ ) processes, which is an extension of the SINAR(1) process of Kachour and Truquet (2011) (see Section 3.4) to the bivariate case. A bivariate process $(X_{t})$ is said to be a B-SINAR $(1)$ process if $X_{t}$ admits the representation

X_{t} = F * X_{t - 1} + Z_{t} \equiv [\begin{matrix} F_{11} & F_{12} \\ F_{21} & F_{22} \end{matrix}] * [\begin{matrix} X_{1, t - 1} \\ X_{2, t - 1} \end{matrix}] + [\begin{matrix} Z_{1, t} \\ Z_{2, t} \end{matrix}], t \in ℤ .

The case

F_{12} * = F_{21} * = 0

can be seen as an extension of Pedeli and Karlis’ model in 4.2 on

ℤ^{2}

. For this case, the authors assume that

Z_{t}

is modeled through a bivariate Skellam distribution Bulla et al. (2014).

5 Discussion

This article has presented a comprehensive review of the literature on thinning-based models for the analysis of univariate and multivariate integer-valued time series with finite and infinite support. We surveyed a wide range of thinning operators which, by construction, are useful for fitting integer-valued time series exhibiting features such as lack of stationarity, positive and negative autocorrelations, and also overdispersion. There are a number of further possibilities for future research in this area. We briefly highlight some of them.

Univariate models: Count models with long memory (in the covariance sense) need to be developed and studied in detail; examples of time series of counts exhibiting long memory can be found in stock transactions Brännäs and Quoreshi (2010); Quoreshi (2014). To develop schemes for detecting possible structural breaks in non-stationary integer-valued time series models is also an impending problem; here Davis et al. (2008) and Kashikar et al. (2013) provide a good starting point. Extensions of INARMA models to space-time INARMA models will be also highly desirable. This topic has been long overlooked in the literature although it has enumerable applications (e.g., number of arrivals to the emergency services at several hospital in the same area). Generally, the areas of $ℤ$ -valued times series and finite-valued time series are still in their infancy.

Multivariate models: Though still facing many challenges, multivariate integer-valued time series modelling has attracted more attention in the recent years. Scotto et al. (2014) suggested to extend the INAR $(1)$ model 2.1 for the analysis of bivariate time series with an infinite range of counts by using the bivariate binomial thinning operator 4.5, i.e., $X_{t} = α \otimes X_{t - 1} + Z_{t}, t \in ℤ .$ This model includes the bivariate models of Karlis and Pedeli (2013) and Pedeli and Karlis (2011); Pedeli and Karlis(2013a); Pedeli and Karlis(2013b) as a special case but allows also for negative cross-covariance. Extensions of multivariate models to higher orders need to be considered, possibly by adapting the approach of Weiß(2008c). Moreover, periodic correlated bivariate and multivariate count models need to be developed. Potential applications of such models can be found in the analysis of fire activity.

Extreme value theory: An important topic for future work is to characterize the tail behaviour and the extremal properties of thinning-based models, and to look at large deviations of partial sums obtained from such models. Anderson (1970) gave a remarkable contribution by defining a particular class of discrete distributions for which the maximum term (under an i.i.d. setting) possesses an almost stable behaviour; extensions for certain stationary sequences were proposed by McCormick and Park (1992) and Hall (1996). The analysis of the extremal behaviour INMA-type models driven by innovations with distribution belonging either to Anderson's class and to the class of heavy-tailed distribution became a topic of lively research; details can be found in Turkman et al. (2014). However, much of the available literature on this topic has been restricted to models based on the binomial thinning operator. Besides analyzing the effect of other thinning operators, also extensions to characterize the extremes of INBL and SETINAR models will be very welcome, as well as the study of the tail behaviour and the large deviation of partial sums based on univariate and bivariate INMA and INBL models.

Footnotes

Acknowledgments

The authors thank the associate editor and the three referees for carefully reading the manuscript and for their valuable comments, which greatly improved the article. This work was supported by the European Regional Development Fund (FEDER) through the COMPETE programme and by the Portuguese government through the FCT, Fundação para a Ciência e a Tecnologia, in the scope of the project UID/MAT/ 04106/2013 (Centro de I&D em Matemática e Aplicações, cidma.mat.ua.pt/) and projects PEst-OE/EEI/UI0127/2014 and UID/CEC/00127/2013 (Instituto de Engenharia Electrónica e Informática de Aveiro, IEETA/UA, www.ieeta.pt). S. Gouveia acknowledges the postdoctoral grant by FCT (ref. SFRH/BPD/87037/2012). The authors are grateful to Dr Xanthi Pedeli, Athens University of Economics and Business (Greece) for contributing the accidents counts data for a,b.

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