In this section, we discuss the construction of PGF estimates for two important classes of stationary NNIV time series. First, using a very general form of the so-called power series (PS) distributions, the estimation procedure of independent identical distributed (IID) time series was examined. After that, using the obtained results, PGF estimators of integer-valued autoregressive (INAR) time series are considered.
3.1. Estimation of IID Time Series
The simplest case of the stationary NNIV time series are the IID series, which we denote as
,
. Similar to that in Stojanović et al. [
20], and Bourguignon and Vasconcellos [
23], we say that the series
has a
PS distribution if its PMF is given as follows:
Here,
is the so-called discrete support of the RV
and
is a function that depends (only) on ;
is a (one-dimensional) unknown parameter;
is a function on , so that when , .
The expression given by Equation (
14) represents the so-called
PS distributions that, for some special choices of
,
and
, allow for obtaining the most of the known distributions (see
Table 1, below). In addition, according to assumption
, it is obvious that, in fact, the power series
converges on the interval
. However, a common assumption is
, so we say that power series
converges on
. Accordingly, the function
is positive and increasing at this interval. Moreover, for an arbitrary
, we denote by
the PGF (of the first order) of RVs
. Thus, according to Equation (
14), one obtains
where the sum above converges when
. Note that Equation (
15) allows for a simple computation of first-order PGFs for some of the well-known NNIV distributions, which are also given in
Table 1.
Furthermore, let
be the appropriate first-order empirical PGF of series
, obtained by its realization
. Thus, the objective function can be written as
where
is the weight function. Finally, in this case, the minimization Equation (
5) reads as follows
As an illustration,
Figure 1 shows plots of the PGFs of some typical NNIV IID time series, along with their empirical PGFs.
In the following, we assume that , so the set will be a compact. Thus, assuming that other regularity conditions in Theorem 1 are fulfilled, we can apply the PGF method. For this purpose, we have taken two different types of distribution of the series . Firstly, we assume that is the IID series with Poisson distribution and, later, that it has a geometric distribution. In the both cases, for estimating the parameter , we generated independent Monte Carlo replications of the series , of the length . Note that this is close to the length of actual series of COVID-19 death cases, which will be considered in the following. To examine the convergence, i.e., the quality of the PGF estimators, the appropriate estimation errors will also be investigated.
The PGF estimates are computed according to a minimization of the integral in Equation (
16). Note that it can be numerically approximated by using an
N-point quadrature formula:
where
are the quadrature nodes and
,
are the corresponding weight coefficients. In our simulation study, quadrature formulas based on Gegenbauer orthogonal polynomials, with weights
and
nodes, were used. More specifically, we consider Gaussian quadratures based on three well-known special cases of Gegenbauer polynomials, i.e., the appropriate weights:
Chebyshev polynomials (of the first kind): ;
Legendre polynomials: ;
Chebyshev polynomials (of the second kind): .
The construction of all these quadrature formulas was realized using the Wolfram Mathematica package “OrthogonalPolinomials”, authored by Cvetković and Milovanović [
24]. After that, the objective function in Equation (
17) is minimized by the “R” procedure “optimize”, based on the Brent minimization algorithm [
25] and realized by the authors’ original codes in “R”.
Summary statistics of the PGF estimates
,
, obtained with the appropriate weights
and computed via aforementioned estimation procedure, are presented in
Table 2. More precisely, it contains the mean values (Mean), minima (Min.), maxima (Max.), and the mean squared estimating errors (MSEE) of obtained PGF estimates of the IID series with Poisson and geometric distribution. (These two distributions are taken as an illustration of the application of the PGF method, that is, to describe the dynamics of actual data in
Section 4, below). The values of objective functions
,
, as the reference estimation errors, are also shown. According to presented results, it is obvious that PGF estimates of both IID series are generally the efficient parameter estimators and have very similar properties. Slightly lower estimation errors are observed in the case of the Gauss–Chebyshev quadrature of the second type, i.e., the weight function
. This is expected, since it is known that this weight forces the points around the coordinate origin and ignores those near the ends of the interval
. On the other hand, the somewhat larger estimation error in the case of Gauss–Chebyshev quadrature of the first kind is a consequence of ‘forcing’ the ends of the interval
, where the weight is infinite. In addition, the results of AN testing of the obtained PGF estimates are also presented in
Table 2. For this purpose, the Anderson–Darling normality test was used and its test statistic (labelled as AD), along with the corresponding
p-values, were calculated using the procedure from the R-package “nortest”, authored by Gross [
26]. According to the values thus obtained, it is evident that the AN property is valid for all PGF estimators, which is in accordance with the previous theoretical results, i.e., the statement of Theorem 1.
3.2. Estimation of INAR Time Series
In this part of our work, using the previous assumptions, we define
the integer-valued autoregressive process (of the first order) or, simply,
INAR(1) process ,
by the recurrence relation
Here,
is the IID series of RVs commonly called
innovations,
is an unknown parameter, and
is the so-called
binomial thinning operator (for more details, see [
27]). More precisely,
X is an arbitrary NNIV RV such that, for any
, the RVs
are mutually independent (also independent of
X) with Bernoulli’s distribution
In that way, it is simply shown that the INAR(1) process
is stationary for each
, with the autocorrelation function (ACF)
As an illustration,
Figure 2 presents sample frequency distributions of the INAR(1) process
, along with their corresponding innovations
, for Poisson and geometric distributions. Furthermore, it will be of interest to use the first-order PGF of the RV
. For that purpose, we prove the following simple statement:
Lemma 1. For an arbitrary and NNIV RV X, the first-order PGF of the RV reads as follows: Proof. According to the definition of the first-order PGF, Equation (
19) and the law of total expectation (see, e.g., Theorem 34.4 in [
28]), we have
Here,
is the first-order PGF of the RVs
. Hence, substituting this expression in Equation (
22) obviously gives Equation (
21). □
Let us also note that some of the important properties of the INAR(1) process can be found in many studies, cf. [
29,
30,
31,
32,
33,
34,
35,
36]. This kind of stochastic model was first introduced in the pioneer work of Al-Osh and Alzaid [
37]. Here, among others, the first order PGF of the INAR(1) process was derived. In the case of PS-distributed innovations
, we generalized this result in the following way:
Theorem 2. Let us assume that the innovations of the INAR(1) process have the PS distribution given by Equation (14). Additionally, suppose that the function has a finite derivative, uniformly bounded on a finite interval . Then, an arbitrary INAR (1) series , given by Equation (19), has a PGF of the first order:as well as PGFs of order : Proof. According to a definition of the PS-distributed RVs
, given by Equation (
14), as well as their first-order PGF, given by Equation (
15), the mathematical expectation of the series
can be obtained as follows:
Based on this and the assumptions of the theorem, for some constant
and any
, we have
where the sum above converges uniformly on
. On the other hand, the sequence
,
is monotone and bounded, so Abel’s convergence criterion implies that, when uniformly on
, the following is valid:
As it is known from Alzaid and Al-Osh [
38], the inequality (
25) represents a necessary and sufficient condition for the infinite-order integer moving average (INMA) representation of the series
:
where the sum above converges almost surely. According to Equation (
26) and the previous Lemma, we have the following:
and the product above converges absolutely at least for all
. Hence, substituting Equation (
15) in Equation (
27), we have Equation (
23). In order to prove the second part of the theorem, notice that, according to the definition (
18) of the series
, for an arbitrary
, the following holds:
Substituting
into Equation (
28) and after some computation similar to the previous one, an explicit expression of the PGF in Equation (
24) can be easily obtained. □
Remark 1. Note that the distribution of the INAR(1) process depends on two unknown parameters . Therefore, in order to estimate them, it is of interest to determine the two-dimensional PGF of , which is simply obtained by replacing in Equation (24):In the following, similar to the case of IID series, the procedure for estimating the parameters of the INAR(1) model with Poisson and geometric innovations will be described. Using Poisson distribution for series , as well as Equations (23) and (29), the one- and two-dimensional PGFs of the INAR(1) process , after some simple computations, can be obtained as follows: In a similar way, for the appropriate PGFs of the INAR(1) process with geometric innovations, one obtainsThree-dimensional plots of the theoretical and empirical PGFs of INAR(1) processes with Poisson and geometric innovations, respectively, are shown in Figure 3. Therefore, notice that the PGFs of the INAR(1) process with geometric innovations are not in closed form. However, they depend on alternating sums and can be easily approximated by finite k-term sums. Thus, for instance, in the case of a one-dimensional PGF, the approximation error amounts to Similar to that in the previous estimation procedure, the PGF estimates
of unknown parameters
are computed according to a minimization of the double integral
with respect to some weight function
. Similar to earlier, the integral in (
30) can be numerically approximated by using some
N-point cubature formula:
where
are the cubature nodes, and
are the corresponding weight coefficients. In our simulations study, we used the same kind of (two-dimensional) weights as in the IID case, i.e., the Gaussian qubature formulas based on Gegenbauer orthogonal polynomials, with weights
and
nodes. After that, the objective function (
30) is minimized by “R” procedure for linearly constrained minimization “constrOptim”, based on the Nelder–Mead optimization method [
39].
As the initial values of this estimation procedure, we used the Yule–Walker (YW) estimates, obtained by solving the equations
Here,
are, respectively, the mean and first autocorrelation of the INAR(1) series
, while
are their sample estimators. Thus, in the case of Poisson distributed innovations
, the YW estimators are obtained as follows:
Similarly, for geometric distributed innovations
, the YW estimates are
Summary statistics of the PGF estimates
,
, obtained using the aforementioned numerical estimation procedure with the appropriate weights
,
, are presented in the following
Table 3. Similar to in the IID case, two different types of innovations
, with Poisson and geometric distributions, were used. Thus, in both cases, the unknown parameters
were estimated through the set of
independent Monte Carlo simulations of the innovation series
and INAR(1) process
of the length
.
In the same way as before,
Table 3 contains the mean values (Mean), minima (Min.), maxima (Max.), as well as the mean squared estimating errors (MSEE) and the objective function
for all obtained PGF estimates. Notice that the thus obtained PGF estimators have very similar properties as in the one-dimensional case, with somewhat higher estimation errors, which is expected. Namely, they are computed using a two-step procedure, using the YW estimated values as the initial ones.
Similar to earlier, the summarized values of
test statistics, along with the appropriate
p-values, are also shown in
Table 3. It can be seen that the AN property is confirmed in almost all cases, that is, for most of the PGF estimators of the parameters
, as well as the corresponding weight functions
. We assume, in the same way as it is shown in Stojanović et al. [
40], that the variations that occur in some simulations, at the significant level of
, are also reasonable as a consequence of this specific, two-stage estimation procedure.