of order p (PINAR(p)S)
Abstract
This paper introduces a new class of models for S-periodically autocorrelated count time series, which has autoregressive structure of order p. This model is an extension of the PINAR(1)S
model. Statistical properties of the model such as mean, variance, marginal and joint distribu- tions are discussed. Moments-based, conditional least squares and quasi-maximum likelihood estimation methods of the parameters are studied and their performances are investigated through Monte Carlo simulations. Under some assumptions, the estimators are asymptotically Normal distributed with rate of convergence of n1/2, where n is the sample size of each sea-
son. The performance of the estimator was investigated for small sample size and the empirical results indicated that the method presented accurate estimates. The model well adjusted the series daily number of medicine dispensing for the treatment of respiratory disease.
Keywords: INAR, Periodic Stationarity, PINAR, Moment-based estimators, Conditional Least Squares, Conditional Maximum Likelihood Estimation.
1
Introduction
The integer autoregressive (INAR) models, initially introduced by the INAR(1) model in Al- Osh & Alzaid (1987), appears as an alternative to the well-known Poisson model family for modeling count time series, see, e.g., Fokianos et al. (2009). These models are based on the thinning operator, see Steutel & Van Harn (1979). In this article, the thinning operator is based on Bernoulli distribution, called binomial thinning operator. The binomial thinning operator ◦ applied on a random variable (r.v.) Y is defined as
α◦ Y =
Y
X
i=1
Ui(α), (4.1)
where Y is a Z+-valued r.v., α ∈ [0, 1] and {Ui(α)}i∈Z+ is a sequence of independent identically
distributed (i.i.d.) r.v.’s which are Bernoulli distributed with parameter α. We assume that the sequence {Ui(α)}i∈Z+ is mutually independent of Y . Note that the empty sum is set to 0 if
Y = 0. The sequence {Ui(α)}i∈Z+ is called a counting sequence. Observe that the probability
of success in the thinning is P(Ui(α) = 1) = α and, conditionally on Y , α ◦ Y ∼ Bin(Y, α).
Further details about thinning based count time series models are given by Scotto et al. (2015) for the univariate and Latour (1997) in the multivariate case, respectively.
An extension of the INAR(1) model that account the p-th order autoregressive structure is the INAR(p), introduced by Alzaid & Al-Osh (1990) and, independently by Du & Li (1991). A discrete time non-negative integer-valued stochastic process {Yt}t∈Z, is said to be an INAR(p) process
if it satisfies the following equation,
Yt= α1◦ Yt−1+· · · + αp◦ Yt−p+ εt,
where 0 ≤ αi < 1 for i = 1, . . . , p − 1 and 0 < αp < 1, {εt} is a sequence of independent
and identically distributed (IID) non-negative integer-valued random variables with finite mean and variance. Alzaid & Al-Osh (1990) presented a model for count time series that has a correlation structure similar to the correlation structure of a conventional ARMA(p, p − 1) for continuous data. We introduce a model based on an extension of the INAR(p) presented by Du & Li (1991), which model is based on a process with a correlation structure identical to the correlation structure of a standard AR(p).
In spite of its flexibility in dealing with higher order autoregressive processes, the INAR(p) model do not account the periodic phenomenon which is quite common in many area of appli- cation. Time series with periodically varying mean, variance and covariance, were introduced by Gladyshev (1961) and are usually called periodically correlated processes (PC). The oc- currence of PC processes in time series is corroborated by real applications in many practical situations, see, e.g., Gardner et al. (2006). Basawa & Lund (2001) studied the asymptotic prop- erties of parameter estimates for specifics periodic autoregressive moving-average (PARMA) models among others, and, recently, Sarnaglia et al. (2010) and Solci et al. (2018) presented robust estimation methods for periodic autoregressive processes (PAR) with application in air pollution data. Even though there are in the literature many studies that focus on periodically correlated processes, the vast majority are dedicated to the analysis and the applications for discrete parameter processes (see Priestley (1981), Definition 3.2), with the application of the PARMA model. However, not much attention has been paid to the analysis of periodically cor- related count series, for example, Monteiro et al. (2010) and Mori˜na et al. (2011). In the former paper, the authors introduced the PINAR(1) model and addressed some statistical properties of the parameter estimators together with some empirical investigation. However, the paper does explore the model in a practical problem. The later paper presents a model based on two-order integer-valued autoregressive time series to analyze the number of hospital emergency service arrivals caused by diseases that present seasonal behavior. The first-order seasonal structure INAR was introduced by Bourguignon et al. (2016) and the class of subset INAR models will be investigated in the forthcoming paper Bondon et al. (2018). The PINAR(1, 1S) model is a
particular case of the PINAR(p)S,
In the remainder of this paper, let N, Z, Z+, R, R+ and C denote the set of positive integers,
integers, non-negative integers, real numbers, non-negative real numbers and complex num- bers, respectively. The integer part of x ∈ R is denoted by bxc and the modulus of n ∈ N with respect to S ∈ Z+ is defined as n MOD S = n − Sbn/Sc. Let {n}S = S, if n MOD S = 0 and
{n}S = nMOD S, otherwise. Clearly, {n}S ∈ {1, . . . , S} for all n ∈ N. Let {ei}i=1,...,d be the
standard basis in Rd, i.e., (e
For all d ∈ N let 1d = (1, . . . , 1)> ∈ Ndand let us denote by Idthe d × d identity matrix. If it is
clear from the context, then we omit the subscript d. Bin(n, α) denotes a binomial distribution with parameters n ∈ N and α ∈ (0, 1); Poi(λ) denotes a Poisson distribution with mean param- eter λ ∈ R+; Geo(q) denotes a Geometric distribution over Z+ with parameter q ∈ (0, 1) and
mean (1 − q)/q. Let E(·) and E(·|·) represent the expectation and the conditional expectation, respectively. Random variables are all defined on a common probability space (Ω, A, P). The organization of the paper is as follows. Section 2 introduces the proposed model, presents the mean and the autocorrelation of the process and some probabilistic properties of the model. Section 3 discuss estimation methods of the parameters, namely the Yule-Walker (moment- based) estimator, the conditional least squares and the quasi-maximum likelihood framework and an alternative estimation procedure. Section 4 presents the simulation and its results, real data application is presented in the Section 5, finally conclusions and final comments are presented at the last section. The appendix shows some proofs and equations mentioned in this article.