First-order 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.

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.

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.

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.

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.

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.

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.

Modeling Medical Data by Flexible Integer-Valued AR(1) Process with Zero-and-One-Inflated Geometric Innovations.

In this paper, we introduce a new stationary first-order integer-valued autoregressive process (INAR) with zero-and-one-inflated geometric innovations that is useful for modeling medical practical data. Basic probabilistic and statistical properties of the model are discussed. Conditional least squares and maximum likelihood estimators are proposed to estimate the model parameters. The performance of the estimation methods is assessed by some Monte Carlo simulation experiments. The zero-and-one-inflated INAR process is subsequently applied to analyze two medical series that include the number of new COVID-19-infected series from Barbados and Poliomyelitis data. The proposed model is compared with other popular competing zero-inflated and zero-and-one-inflated INAR models on the basis of some goodness-of-fit statistics and selection criteria, where it shows to provide better fitting and hence can be considered as another important commendable model in the class of INAR models.

On Discrete Poisson–Mirra Distribution: Regression, INAR(1) Process and Applications

Several pieces of research have spotlighted the importance of count data modelling and its applications in real-world phenomena. In light of this, a novel two-parameter compound-Poisson distribution is developed in this paper. Its mathematical functionalities are investigated. The two unknown parameters are estimated using both maximum likelihood and Bayesian approaches. We also offer a parametric regression model for the count datasets based on the proposed distribution. Furthermore, the first-order integer-valued autoregressive process, or INAR(1) process, is also used to demonstrate the utility of the suggested distribution in time series analysis. The unknown parameters of the proposed INAR(1) model are estimated using the conditional maximum likelihood, conditional least squares, and Yule–Walker techniques. Simulation studies for the suggested distribution and the INAR(1) model based on this innovative distribution are also undertaken as an assessment of the long-term performance of the estimators. Finally, we utilized three real datasets to depict the new model’s real-world applicability.

Discrete Pseudo Lindley Distribution: Properties, Estimation and Application on INAR(1) Process

In this paper, we introduce a discrete version of the Pseudo Lindley (PsL) distribution, namely, the discrete Pseudo Lindley (DPsL) distribution, and systematically study its mathematical properties. Explicit forms gathered for the properties such as the probability generating function, moments, skewness, kurtosis and stress–strength reliability made the distribution favourable. Two different methods are considered for the estimation of unknown parameters and, hence, compared with a broad simulation study. The practicality of the proposed distribution is illustrated in the first-order integer-valued autoregressive process. Its empirical importance is proved through three real datasets.

Modelling and monitoring of INAR(1) process with geometrically inflated Poisson innovations

ABSTRACT To analyse count time series data inflated at the r + 1 values , we propose a new first-order integer-valued autoregressive process with r-geometrically inflated Poisson innovations. Some statistical properties together with conditional maximum likelihood estimate are provided. For the purpose of statistical monitoring, we focus on the cumulative sum chart, exponentially weighted moving average chart and combined jumps chart towards the proposed process. Numerical simulations indicate that the conditional maximum likelihood estimator is unbiased. Moreover, the cumulative sum chart is the best choice to monitor our model in practice. Some applications about telephone complaints data are provided to illustrate the proposed methods.

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.

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.

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 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.

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.

Some estimation and forecasting procedures in Possion-Lindley INAR(1) process

We study the estimation and forecasting in first-order integer-valued autoregressive process with Poisson-Lindley (PLINAR(1)) marginal distribution (Mohammadpour et al.). Quasi-likelihood estimators are proposed for the parameters of interest and their asymptotic properties are derived. Two methods for coherent point prediction are given and the prediction intervals for future data are constructed. We present some simulations to verify rationality of the proposed estimation and prediction methods. An application to a real data about animal’s anorexia is also provided.

Extended Poisson INAR(1) processes with equidispersion, underdispersion and overdispersion

ABSTRACTReal count data time series often show the phenomenon of the underdispersion and overdispersion. In this paper, we develop two extensions of the first-order integer-valued autoregressive process with Poisson innovations, based on binomial thinning, for modeling integer-valued time series with equidispersion, underdispersion, and overdispersion. The main properties of the models are derived. The methods of conditional maximum likelihood, Yule–Walker, and conditional least squares are used for estimating the parameters, and their asymptotic properties are established. We also use a test based on our processes for checking if the count time series considered is overdispersed or underdispersed. The proposed models are fitted to time series of the weekly number of syphilis cases and monthly counts of family violence illustrating its capabilities in challenging the overdispersed and underdispersed count data.