Integer-valued Autoregressive Process Research Articles

A new approach to model the counts of earthquakes: INARPQX(1) process

This paper introduces a first-order integer-valued autoregressive process with a new innovation distribution, shortly INARPQX(1) process. A new innovation distribution is obtained by mixing Poisson distribution with quasi-xgamma distribution. The statistical properties and estimation procedure of a new distribution are studied in detail. The parameter estimation of INARPQX(1) process is discussed with two estimation methods: conditional maximum likelihood and Yule-Walker. The proposed INARPQX(1) process is applied to time series of the monthly counts of earthquakes. The empirical results show that INARPQX(1) process is an important process to model over-dispersed time series of counts and can be used to predict the number of earthquakes with a magnitude greater than four.

Conditional least squares estimation for the SINAR(1, 1) process

This paper presents the conditional least squares (CLS) estimation procedure for the class of first-order spatial integer-valued autoregressive processes, denote by SINAR(1, 1). We derive the asymptotic properties of the CLS estimators of model parameters. Simulation results are presented to assess the performance of estimators under finite sample sizes and under equidispersion and overdispersion. To illustrate the proposed methodology, two grid sampled count data sets are analyzed.

A first-order integer-valued autoregressive process with zero-modified Poisson-Lindley distributed innovations

In this paper, we introduce a first-order integer-valued autoregressive process with zero-modified Poisson-Lindley distributed innovations based on the binomial thinning operator. Some statistical and conditional properties of the model are presented, several estimation methods are applied to estimate the unknown parameters and, for the Yule-Walker and conditional least square estimators, the asymptotic distributions are obtained. Besides, a simulation study is conducted to compare the performance of the proposed estimators and, finally, two real count time series data are applied to demonstrate the usefulness of the proposed model.

Random coefficients integer-valued threshold autoregressive processes driven by logistic regression

In this article, we introduce a new random coefficients self-exciting threshold integer-valued autoregressive process. The autoregressive coefficients are driven by a logistic regression structure, so that the explanatory variables can be included. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares and conditional maximum likelihood estimators, as well as the asymptotic properties of the estimators, are discussed. The nonlinearity test of the model and existence test of explanatory variables are also addressed. As an illustration, we evaluate our estimates through a simulation study. Finally, we apply our method to the data sets of sexual offences in Ballina, New South Wales (NSW), Australia, with two covariates of temperature and drug offences. The result reveals that the proposed model fits the data sets well.

First-order integer-valued autoregressive process with Markov-switching coefficients

A class of integer-valued autoregressive models is considered, which are based on a latten process e.g., a Markov chain. Some statistical properties of this class of models are discussed. Moreover, parameter estimation and forecasting are addressed. A Monte Carlo simulation study is conducted to examine the finite sample performance of the given estimation procedure. Finally, the performance of these models is illustrated by empirical applications to two sets of real counts.

A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models

Abstract This study introduces the Poisson-Bilal distribution and its associated two models for modeling the over-dispersed count data sets. The Poisson-Bilal distribution has tractable properties and explicit forms for its statistical properties. A new over-dispersed count regression model and integer-valued autoregressive process with flexible innovation distribution are defined and studied comprehensively. Two real data sets are analyzed to prove empirically the importance of proposed models. Empirical findings show that the Poisson-Bilal distribution has important application fields in time series and regression modeling.

Bayesian analysis of the p-order integer-valued AR process with zero-inflated Poisson innovations

In recent years, there has been a considerable interest to study count time series with a dependence structure and appearance of excess of zeros values. Such series are commonly encountered in diverse disciplines, such as economics, financial research, environmental science, public health, among others. In this paper, we propose a stationary p-order integer-valued autoregressive process with zero-inflated Poisson innovations, called the ZINAR(p) times series model. We study some of its theoretical properties and develop a Markov chain Monte Carlo (MCMC) algorithm for inferring parameters from Bayesian perspectives. Finally, we demonstrate the utility of proposed ZINAR(p) model through simulated and real data examples.

A perturbation analysis of Markov chains models with time-varying parameters

We study some regularity properties in locally stationary Markov models which are fundamental for controlling the bias of nonparametric kernel estimators. In particular, we provide an alternative to the standard notion of derivative process developed in the literature and that can be used for studying a wide class of Markov processes. To this end, for some families of $V$-geometrically ergodic Markov kernels indexed by a real parameter $u$, we give conditions under which the invariant probability distribution is differentiable with respect to $u$, in the sense of signed measures. Our results also complete the existing literature for the perturbation analysis of Markov chains, in particular when exponential moments are not finite. Our conditions are checked on several original examples of locally stationary processes such as integer-valued autoregressive processes, categorical time series or threshold autoregressive processes.

Forecasting with two generalized integer-valued autoregressive processes of order one in the mutual random environment

In this article, we consider two univariate random environment integer-valued autoregressive processes driven by the same hidden process. A model of this kind is capable of describing two correlated non-stationary counting time series using its marginal variable parameter values. The properties of the model are presented. Some parameter estimators are described and implemented on the simulated time series. The introduction of this bivariate integer-valued autoregressive model with a random environment is justified at the end of the paper, where its real-life data-fitting performance was checked and compared to some other appropriate models. The forecasting properties of the model are tested on a few data sets, and forecasting errors are discussed through the residual analysis of the components that comprise the model.

Modelling heavy-tailedness in count time series

Count data frequently exhibit overdispersion, zero inflation and even heavy-tailedness (the tail probabilities are non-negligible or decrease very slowly) in practical applications. Many models have been proposed for modelling count data with overdispersion and zero inflation, but heavy-tailedness is less considered. The proposed model, a new integer-valued autoregressive process with generalized Poisson-inverse Gaussian innovations, is capable of capturing these features. The generalized Poisson-inverse Gaussian family is very flexible, which includes Poisson distribution, Poisson inverse Gaussian distribution, discrete stable distribution and so on. Stationarity and ergodicity of this model are investigated and the expressions of marginal mean and variance are provided. Conditional maximum likelihood is used for estimating the parameters, and consistency and asymptotic normality for the estimators are presented. Further, we consider the h-step forecast and diagnostics for the proposed model. The proposed model is applied to three real data examples. In the first example, we consider the monthly number of cases of Polio, which validates that the proposed model can take into account count data with excessive zeros. Then, we illustrate the use of the proposed model through an application to the numbers of National Science Foundation fundings. Finally, we apply the proposed model to the numbers of transactions in 5-min intervals for the stock traded at Empire District Electric Company. The second and third examples show that the proposed model has a good performance in modelling heavy-tailed count data.

On shifted integer-valued autoregressive model for count time series showing equidispersion, underdispersion or overdispersion

In this paper, we introduce a first order integer-valued autoregressive process with Borel innovations based on the binomial thinning. This model is suitable to modeling zero truncated count time series with equidispersion, underdispersion and overdispersion. The basic properties of the process are obtained. To estimate the unknown parameters, the Yule-Walker (YW), conditional least squares (CLS) and conditional maximum likelihood (CML) methods are considered. The asymptotic distribution of CLS estimators are obtained and hypothesis tests to test an equidispersed model against an underdispersed or overdispersed model are formulated. A Monte Carlo simulation is presented analyzing the estimators performance in finite samples. Two applications to real data are presented to show that the Borel INAR(1) model is suited to model underdispersed and overdispersed data counts.

First-order random coefficient INAR process with dependent counting series

In this paper, we propose a first-order random coefficient integer-valued autoregressive process with dependent counting series. Some moments and stationary ergodicity of the process are established. The maximum-likelihood estimators of the parameters of interest are presented. We conduct some simulation studies to assess the performance of our method. An example about crime data is provided for practical application.

A new thinning-based INAR(1) process for underdispersed or overdispersed counts

Underdispersed and overdispersed phenomena are often observed in practice. To deal with these phenomena, we introduce a new thinning-based integer-valued autoregressive process. Some probabilistic and statistical properties of the process are obtained. The asymptotic normality of the estimators of the model parameters, using conditional least squares, weighted conditional least squares and modified quasi-likelihood methods, are presented. One overdispersed real-data example and one underdispersed real-data example are given to show the flexibility and superiority of the new model.

A New Three-Parameter Discrete Distribution With Associated INAR(1) Process and Applications

The aim of this article is to propose a new three-parameter discrete Lindley distribution. A wide range of its structural properties are investigated. This includes the shape of the probability mass function, hazard rate function, moments, skewness, kurtosis, index of dispersion, mean residual life, mean past life and stress-strength reliability. These properties are expressed in explicit forms. The maximum likelihood approach is used to estimate the model parameters. A detailed simulation study is carried out to examine the bias and mean square error of the estimators. Using the proposed distribution, a new first-order integer-valued autoregressive process is introduced for the over-dispersed, equi-dispersed and under-dispersed time series of counts. To demonstrate the importance of the proposed distribution, three data sets on coronavirus, length of stay at psychiatric ward and monthly counts of larceny calls are analyzed.

The family of the bivariate integer-valued autoregressive process (BINAR(1)) with Poisson–Lindley (PL) innovations

ABSTRACTIn the literature of discrete-valued time series modelling, various bivariate integer-valued autoregressive time series models of order 1 (BINAR(1)) have been proposed particularly based on the binomial thinning mechanism and with different innovation distributions. These BINAR(1)s are mostly suitable for modelling bivariate counting series of varied levels of overdispersion. Recently, in the context of overdispersion, the INAR(1) with Poisson–Lindley (PL) innovations have been introduced to provide superior model fitness criteria than other competing INAR(1)s. Thus, this paper proposes to develop classes of BINAR(1)PL, including BINAR(1)PL(I) and BINAR(1)PL(II), under different cross-correlation functions. The parameter estimations are conducted via the conditional maximum likelihood (CML) approach. Monte Carlo simulation experiments are implemented to assess the asymptotic properties of the CML estimators under different combinations of the cross-correlation parameters. The proposed models are also applied to the Pittsburg crimes data and compared with other competing popular overdispersed BINAR(1) models.

An integer-valued autoregressive process for seasonality

ABSTRACTIn this paper, we propose an integer-valued autoregressive process of order 1 for seasonality with period d and intra-seasonally dependent innovations (). Model properties are provided for the univariate and multivariate representation of the process. A computationally fast estimation method, which is based on conditional least squares with parameter restrictions, is proposed for the multivariate model representation and compared with a likelihood-based estimation method via the Monte Carlo simulation. An empirical application for different types of Chicago crime data is carried out in order to assess whether the proposed model is able to capture adequately the seasonality patterns in non-synthetic data.

CONDITIONAL LEAST SQUARES ESTIMATION OF THE PARAMETERS OF HIGHER ORDER RANDOM ENVIRONMENT INAR MODElS

Two different random environment INAR models of higher order, precisely RrNGINARmax(p) and RrNGINAR1(p), are presented as a new approach to modeling non-stationary nonnegative integer-valued autoregressive processes. The interpretation of these models is given in order to better understand the circumstances of their application to random environment counting processes. The estimation statistics, defined using the Conditional Least Squares (CLS) method, is introduced and the properties are tested on the replicated simulated data obtained by RrNGINAR models with different parameter values. The obtained CLS estimates are presented and discussed.

Control charts based on dependent count data with deflation or inflation of zeros

ABSTRACTThe process of serially dependent counts with deflation or inflation of zeros is commonly observed in many applications. This paper investigates the monitoring of such a process, the first-order zero-modified geometric integer-valued autoregressive process (ZMGINAR(1)). In particular, two control charts, the upper-sided and lower-sided CUSUM charts, are developed to detect the shifts in the mean process of the ZMGINAR(1). Both the average run length performance and the standard deviation of the run length performance of these two charts are investigated by using Markov chain approaches. Also, an extensive simulation is conducted to assess the effectiveness or performance of the charts, and the presented methods are applied to two sets of real data arising from a study on the drug use.

Monitoring a bivariate INAR(1) process with application to Hepatitis A

In this article, we study multivariate monitoring systems based on a bivariate integer-valued autoregressive process of order 1, BINAR(1). The charting procedures are evaluated using extensive simulated shift scenarios, and are compared to multiple univariate charts. The proposed methods improve the efficiency of surveillance systems by taking account of both pairwise correlation and autocorrelation in a bivariate data structure. The methods also account for overdispersion in responses for which the Poisson distribution is inappropriate. An example of hepatitis A incidents in Australia is presented to demonstrate the application of these methods. The proposed methods have important applications in public healthcare and biosurveillance as well as industrial and business applications.

A New Generalization of Geometric Distribution with Properties and Applications

In this study, a new two-parameter mixed-Poisson distribution is proposed. Statistical properties of the proposed distribution are studied comprehensively. The maximum likelihood estimation method is used to estimate unknown model parameters. A simulation study is conducted to evaluate the asymptotic efficiencies of the maximum likelihood estimators of model parameters. The usefulness of proposed distribution is demonstrated in first-order integer-valued autoregressive process, shortly INAR(1). Empirical findings show that the proposed INAR(1) process provides better results than other competitive models when the time series of counts display over-dispersion.