Integer-valued Autoregressive Process Research Articles

Randomness Test of Thinning Parameters for the NBRCINAR(1) Process

Non-negative integer-valued time series are usually encountered in practice, and a variety of integer-valued autoregressive processes based on various thinning operators are commonly used to model these count data with temporal dependence. In this paper, we consider a first-order integer-valued autoregressive process constructed by the negative binomial thinning operator with random coefficients, to address the problem of constant thinning parameters which might not always accurately represent real-world settings because of numerous external and internal causes. We estimate the model parameters of interest by the two-step conditional least squares method, obtain the asymptotic behaviors of the estimators, and furthermore devise a technique to test the constancy of the thinning parameters, which is essential for determining whether or not the proposed model should consider the parameters’ randomness. The effectiveness and dependability of the suggested approach are illustrated by a series of thorough simulation studies. Finally, two real-world data analysis examples reveal that the suggested approach is very useful and flexible for applications.

INAR(1) process with weighted negative binomial Lindley distributed innovations and applications to criminal and COVID-19 data

In this study, we introduce a pliant stationary first-order integer-valued autoregressive (INAR) process with weighted negative binomial Lindley innovations. The main properties of the model are derived. The methods of conditional maximum likelihood, conditional least square and Yule-Walker are used for estimating the process parameters, while the efficiency of these three methods is evaluated through a simulation study. Finally, the practical aspect of the proposed INAR(1) process is discussed on two time series of the monthly number of criminal mischief reports in Pittsburgh and the daily COVID-19 death cases in Paraguay.

Bivariate Poisson Generalized Lindley Distributions and the Associated BINAR(1) Processes

This paper proposes new bivariate distributions based on the Poisson generalized Lindley distribution as marginal. These models include the basic bivariate Poisson generalized Lindley (BPGL) and the Sarmanov-based bivariate Poisson generalized Lindley (SPGL) distributions. Subsequently, we introduce the BPGL and SPGL distributions as joint innovation distributions in a novel bivariate first-order integer-valued autoregressive process (BINAR(1)) based on binomial thinning. The model parameters in the BPGL and SPGL distributions are estimated using the method of maximum likelihood (ML) while we apply the conditional maximum likelihood (CML) for the BINAR(1) process. We conduct some simulation experiments to assess the small and large sample performances. Further, we implement the new BINAR(1)s to the Pittsburgh crime series data and they show better fitting criteria than other competing BINAR(1) models in the literature.

Change-points analysis for generalized integer-valued autoregressive model via minimum description length principle

This article considers the problem of modeling a class of count time series with multiple change-points using segmented generalized integer-valued autoregressive (S-GINAR) processes. The minimum description length principle (MDL) is applied to study the statistical inference for the S-GINAR model, and the consistency results of the MDL model selection procedure are established respectively under the condition of known and unknown number of change-points. To find the “best” combination of the number of change-points, the locations of change-points, the order of each segment and its parameters, a genetic algorithm with simulated annealing is implemented to solve this difficult optimization problem. In particular, the simulated annealing process makes up for the precocious problem of the traditional genetic algorithm. Numerical results from simulation experiments and three examples of real data analysis show that the procedure has excellent empirical properties.

Multivariate threshold integer-valued autoregressive processes with explanatory variables

To capture the multivariate count time series showing piecewise phenomena, we introduce a class of first-order multivariate threshold integer-valued autoregressive process. The component of autoregressive coefficient is driven by explanatory variables via a logistic link function. Basic probabilistic and statistical properties of the model are discussed. The model parameters are estimated by means of conditional least squares and conditional maximum likelihood methods. A new algorithm to estimate the threshold parameter of the model is also provided. Moreover, the nonlinearity test of the model and existence test of explanatory variables are well addressed using a Wald-type test. Forecasting problem and model extension problem are also considered. Finally, some numerical results of the estimates and a real data example are presented.

On the discrete analogue of the Teissier distribution and its associated INAR(1) process

The continuous Teissier distribution was proposed by the French biologist Georges Teissier in 1934. This paper introduces the one-parameter discrete analogue distribution and studies some of its statistical properties, focussing the attention on the computer generation of pseudo-random data and the parameter estimation problem. More precisely, the new discrete distribution is suitable to deal with both underdispersed and overdispersed count data, the quantile function can be expressed in closed form in terms of the Lambert W function and the failure rate function is increasing. Monte Carlo simulation experiments revealed that estimation methods such as maximum likelihood, least squares and quantile least squares may produce estimates far away from the true parameter value. This drawback can be overcome due to the existence of an analytical expression for the quantile function and then accurate estimates are obtained by the maximum likelihood method even for small sample sizes. Additionally, the new distribution is also used to derive a first-order integer-valued autoregressive process. Different real data sets are used to illustrate that the proposed distribution provides a better fit than other alternative models and also outperforms other competitive processes when time series of counts are considered.

New bivariate Poisson extended exponential distributions and associated BINAR(1) processes with applications

This study proposes two types of bivariate Poisson extended exponential distributions: the basic bivariate Poisson extended exponential distribution and the Sarmanov-based bivariate Poisson extended exponential distribution. The two bivariate Poisson extended exponential distributions are then introduced as joint innovation distributions in a bivariate first-order integer-valued autoregressive process based on binomial thinning. The model parameters are estimated using maximum likelihood in basic bivariate Poisson extended exponential and Sarmanov-based bivariate Poisson extended exponential distributions. Conditional maximum likelihood is applied to the bivariate first-order integer-valued autoregressive process. Simulation experiments are used to evaluate the performance of small and large samples. In addition, the newly developed bivariate first-order integer-valued autoregressive models are then applied to the Pittsburgh crime series data and candies data. We show that they fit better than existing bivariate first-order integer-valued autoregressive models described in the literature.

A New INAR(1) Model for ℤ-Valued Time Series Using the Relative Binomial Thinning Operator

Abstract A new first-order integer-valued autoregressive process (INAR(1)) with extended Poisson innovations is introduced based on a signed version of the thinning operator, called relative binomial thinning operator, which can be considered as an extension of standard binomial thinning operator introduced by Steutel, F.W. and van Harn, K. (1979. Discrete analogues of self-decomposability and stability. Ann. Probab. 7: 893–899). It is appropriate for modeling Z $\mathbb{Z}$ -valued time series and either positive or negative correlations. Some properties of the process are established. Conditional least squares, Yule–Walker and conditional maximum likelihood methods are considered for the parameter estimation of the model. Moreover, simulation experiments are carried out to attest to the performance of the estimation methods. The applicability of the proposed model is investigated through a practical data set of the Saudi stock market.

Zero-and-One Integer-Valued AR(1) Time Series with Power Series Innovations and Probability Generating Function Estimation Approach

Zero-and-one inflated count time series have only recently become the subject of more extensive interest and research. One of the possible approaches is represented by first-order, non-negative, integer-valued autoregressive processes with zero-and-one inflated innovations, abbr. ZOINAR(1) processes, introduced recently, around the year 2020 to the present. This manuscript presents a generalization of ZOINAR processes, given by introducing the zero-and-one inflated power series (ZOIPS) distributions. Thus, the obtained process, named the ZOIPS-INAR(1) process, has been investigated in terms of its basic stochastic properties (e.g., moments, correlation structure and distributional properties). To estimate the parameters of the ZOIPS-INAR(1) model, in addition to the conditional least-squares (CLS) method, a recent estimation technique based on probability-generating functions (PGFs) is discussed. The asymptotic properties of the obtained estimators are also examined, as well as their Monte Carlo simulation study. Finally, as an application of the ZOIPS-INAR(1) model, a dynamic analysis of the number of deaths from the disease COVID-19 in Serbia is considered.

Empirical Likelihood for a First-Order Generalized Random Coefficient Integer-Valued Autoregressive Process.

In this paper, the authors consider the empirical likelihood method for a first-order generalized random coefficient integer-valued autoregressive process. The authors establish the log empirical likelihood ratio statistic and obtain its limiting distribution. Furthermore, the authors investigate the point estimation, confidence regions and hypothesis testing for the parameters of interest. The performance of empirical likelihood method is illustrated by a simulation study and a real data example.

The balanced discrete Burr–Hatke model and mixing INAR(1) process: properties, estimation, forecasting and COVID-19 applications

The main concern of this paper is providing a flexible discrete model that captures every kind of dispersion (equi-, over- and under-dispersion). Based on the balanced discretization method, a new discrete version of Burr–Hatke distribution is introduced with the partial moment-preserving property. Some statistical properties of the new distribution are introduced, and the applicability of proposed model is evaluated by considering counting series. A new integer-valued autoregressive (INAR) process based on the mixing Pegram and binomial thinning operators with discrete Burr–Hatke innovations is introduced, which can model contagious data properly. The different estimation approaches of parameters of the new process are provided and compared through the Monte Carlo simulation scheme. The performance of the proposed process is evaluated by four data sets of the daily death counts of the COVID-19 in Austria, Switzerland, Nigeria and Slovenia in comparison with some competitor INAR(1) models, along with the Pearson residual analysis of the assessing model. The goodness of fit measures affirm the adequacy of the proposed process in modeling all COVID-19 data sets. The fundamental prediction procedures are considered for new process by classic, modified Sieve bootstrap and Bayesian forecasting methods for all COVID-19 data sets, which is concluded that the Bayesian forecasting approach provides more reliable results.

Two Features of the GINAR(1) Process and Their Impact on the Run-Length Performance of Geometric Control Charts.

The geometric first-order integer-valued autoregressive process (GINAR(1)) can be particularly useful to model relevant discrete-valued time series, namely in statistical process control. We resort to stochastic ordering to prove that the GINAR(1) process is a discrete-time Markov chain governed by a totally positive order 2 (TP2) transition matrix.Stochastic ordering is also used to compare transition matrices referring to pairs of GINAR(1) processes with different values of the marginal mean. We assess and illustrate the implications of these two stochastic ordering results, namely on the properties of the run length of geometric charts for monitoring GINAR(1) counts.

Estimation and bootstrap for stochastically monotone Markov processes

The Markov property is shared by several popular models for time series such as autoregressive or integer-valued autoregressive processes as well as integer-valued ARCH processes. A natural assumption which is fulfilled by corresponding parametric versions of these models is that the random variable at time t gets stochastically greater conditioned on the past, as the value of the random variable at time t-1 increases. Then the associated family of conditional distribution functions has a certain monotonicity property which allows us to employ a nonparametric antitonic estimator. This estimator does not involve any tuning parameter which controls the degree of smoothing and is therefore easy to apply. Nevertheless, it is shown that it attains a rate of convergence which is known to be optimal in similar cases. This estimator forms the basis for a new method of bootstrapping Markov chains which inherits the properties of simplicity and consistency from the underlying estimator of the conditional distribution function.

An INAR(1) Time Series Model via a Modified Discrete Burr-Hatke with Medical Applications.

This paper introduces a flexible discrete transmuted record type discrete Burr-Hatke (TRT-DBH) model that seems suitable for handling over-dispersion and equi-dispersion in count data analysis. Further to the elegant properties of the TRT-DBH, we propose, in the time series context, a first-order integer-valued autoregressive process with TRT-DBH distributed innovations [TRBH-INAR(1)]. The moment properties and inferential procedures of this new INAR(1) process are studied. Some Monte Carlo simulation experiments are executed to assess the consistency of the parameters of the TRBH-INAR(1) model. To further motivate its purpose, the TRBH-INAR(1) is applied to analyze the series of the COVID-19 deaths in Netherlands and the series of infected cases due to the Tularaemia disease in Bavaria. The proposed TRBH-INAR(1) model yields superior fitting criteria than other established competitive INAR(1) models in the literature. Further diagnostics related to the residual analysis and forecasting based on the TRBH-INAR(1) model are also discussed. Based on modified Sieve bootstrap predictors, we provide integer forecasts of future death of COVID-19 and infected of Tularemia.

On Periodic Generalized Poisson INAR(p) Models

This paper deals with some probabilistic and statistical properties of periodic generalized Poisson integer-valued autoregressive processes of order p, Necessary and sufficient conditions for the periodic stationarity, both in mean and second order, are established. The closed-forms of the mean and the second moment are, under these conditions, obtained. Moreover, the Wold–Cramér expression of the underlying second-order periodically stationary process is then established. The autocovariance structure is studied, while providing the closed-form of the periodic autocorrelation function. The Yule–Walker (YW), the two-stage conditional least squares (CLS) and the conditional maximum likelihood (CML) estimation methods of the underlying parameters are obtained. An intensive simulation study and an application on a real count data consisting of the daily number of daytime road accidents in Schiphol area, in Netherlands are provided.

The balanced discrete triplet Lindley model and its INAR(1) extension: properties and COVID-19 applications.

This paper proposes a new flexible discrete triplet Lindley model that is constructed from the balanced discretization principle of the extended Lindley distribution. This model has several appealing statistical properties in terms of providing exact and closed form moment expressions and handling all forms of dispersion. Due to these, this paper explores further the usage of the discrete triplet Lindley as an innovation distribution in the simple integer-valued autoregressive process (INAR(1)). This subsequently allows for the modeling of count time series observations. In this context, a novel INAR(1) process is developed under mixed Binomial and the Pegram thinning operators. The model parameters of the INAR(1) process are estimated using the conditional maximum likelihood and Yule-Walker approaches. Some Monte Carlo simulation experiments are executed to assess the consistency of the estimators under the two estimation approaches. Interestingly, the proposed INAR(1) process is applied to analyze the COVID-19 cases and death series of different countries where it yields reliable parameter estimates and suitable forecasts via the modified Sieve bootstrap technique. On the other side, the new INAR(1) with discrete triplet Lindley innovations competes comfortably with other established INAR(1)s in the literature.

Statistical modelling of COVID-19 and drug data via an INAR(1) process with a recent thinning operator and cosine Poisson innovations.

In this paper, we propose the first-order stationary integer-valued autoregressive process with the cosine Poisson innovation, based on the negative binomial thinning operator. It can be equi-dispersed, under-dispersed and over-dispersed. Therefore, it is flexible for modelling integer-valued time series. Some statistical properties of the process are derived. The parameters of the process are estimated by two methods of estimation and the performances of the estimators are evaluated via some simulation studies. Finally, we demonstrate the usefulness of the proposed model by modelling and analyzing some practical count time series data on the daily deaths of COVID-19 and the drug calls data.

Bivariate Poisson 2Sum-Lindley Distributions and the Associated BINAR(1) Processes

Discrete-valued time series modeling has witnessed numerous bivariate first-order integer-valued autoregressive process or BINAR(1) processes based on binomial thinning and different innovation distributions. These BINAR(1) processes are mainly focused on over-dispersion. This paper aims to propose new bivariate distributions and processes based on a recently proposed over-dispersed distribution: the Poisson 2S-Lindley distribution. The new bivariate distributions, denoted by the abbreviations BP2S-L(I) and BP2S-L(II), are then used as innovation distributions for the BINAR(1) process. Properties are investigated for both distributions as well as for the BINAR(1) processes. The distribution parameters are estimated using the maximum likelihood method, and the BINAR(1)BP2S-L(I) and BINAR(1)BP2S-L(II) process parameters are estimated using the conditional least squares and conditional maximum likelihood methods. Monte Carlo simulation experiments are conducted to study large and small sample performances and for the comparison of the estimation methods. The Pittsburgh crime series and candy sales datasets are then used to compare the new BINAR(1) processes to some other existing BINAR(1) processes in the literature.

An empirical-likelihood-based structural-change test for INAR processes

The empirical likelihood ratio (ELR) test is proposed for uncovering a structural change in integer-valued autoregressive (INAR) processes. The limiting distribution is derived under the null hypothesis that the parameter did not change at the anticipated change points. To evaluate the finite-sample performance of the proposed ELR test, the empirical sizes and powers are investigated in a simulation study. The ELR test is also applied to real data on infectious disease and crime counts.

Poisson Extended Exponential Distribution with Associated INAR(1) Process and Applications

The significance of count data modeling and its applications to real-world phenomena have been highlighted in several research studies. The present study focuses on a two-parameter discrete distribution that can be obtained by compounding the Poisson and extended exponential distributions. It has tractable and explicit forms for its statistical properties. The maximum likelihood estimation method is used to estimate the unknown parameters. An extensive simulation study was also performed. In this paper, the significance of the proposed distribution is demonstrated in a count regression model and in a first-order integer-valued autoregressive process, referred to as the INAR(1) process. In addition to this, the empirical importance of the proposed model is proved through three real-data applications, and the empirical findings indicate that the proposed INAR(1) model provides better results than other competitive models for time series of counts that display overdispersion.