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

Some developments on seasonal INAR processes with application to influenza data

Influenza epidemic data are seasonal in nature. Zero-inflation, zero-deflation, overdispersion, and underdispersion are frequently seen in such number of cases of disease (count) data. To explain these counts' features, this paper introduces a flexible model for nonnegative integer-valued time series with a seasonal autoregressive structure. Some probabilistic properties of the model are discussed for general seasonal INAR(p) model and three estimation methods are used to estimate the model parameters for its special case seasonal INAR(1) model. The performance of the estimation procedures has been studied using simulation. The proposed model is applied to analyze weekly influenza data from the Breisgau- Hochschwarzwald county of Baden-Württemberg state, Germany. The empirical findings show that the suggested model performs better than existing models.

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.

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.

Re-visiting the COVID-19 analysis using the class of high ordered integer-valued time series models with harmonic features

The COVID-19 series is obviously one of the most volatile time series with lots of spikes and oscillations. The conventional integer-valued auto-regressive time series models (INAR) may be limited to account for such features in COVID-19 series such as severe over-dispersion, excess of zeros, periodicity, harmonic shapes and oscillations. This paper proposes alternative formulations of the classical INAR process by considering the class of high-ordered INAR models with harmonic innovation distributions. Interestingly, the paper further explores the bivariate extension of these high-ordered INARs. South Africa and Mauritius’ COVID-19 series are re-scrutinized under the optic of these new INAR processes. Some simulation experiments are also executed to validate the new models and their estimation procedures.

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.

Re-analyzing the SARS-CoV-2 series using an extended integer-valued time series models: A situational assessment of the COVID-19 in Mauritius

This paper proposes some high-ordered integer-valued auto-regressive time series process of order p (INAR(p)) with Zero-Inflated and Poisson-mixtures innovation distributions, wherein the predictor functions in these mentioned distributions allow for covariate specification, in particular, time-dependent covariates. The proposed time series structures are tested suitable to model the SARs-CoV-2 series in Mauritius which demonstrates excess zeros and hence significant over-dispersion with non-stationary trend. In addition, the INAR models allow the assessment of possible causes of COVID-19 in Mauritius. The results illustrate that the event of Vaccination and COVID-19 Stringency index are the most influential factors that can reduce the locally acquired COVID-19 cases and ultimately, the associated death cases. Moreover, the INAR(7) with Zero-inflated Negative Binomial innovations provides the best fitting and reliable Root Mean Square Errors, based on some short term forecasts. Undeniably, these information will hugely be useful to Mauritian authorities for implementation of comprehensive policies.

A first-order binomial-mixed Poisson integer-valued autoregressive model with serially dependent innovations

Motivated by the extended Poisson INAR(1), which allows innovations to be serially dependent, we develop a new family of binomial-mixed Poisson INAR(1) (BMP INAR(1)) processes by adding a mixed Poisson component to the innovations of the classical Poisson INAR(1) process. Due to the flexibility of the mixed Poisson component, the model includes a large class of INAR(1) processes with different transition probabilities. Moreover, it can capture some overdispersion features coming from the data while keeping the innovations serially dependent. We discuss its statistical properties, stationarity conditions and transition probabilities for different mixing densities (Exponential, Lindley). Then, we derive the maximum likelihood estimation method and its asymptotic properties for this model. Finally, we demonstrate our approach using a real data example of iceberg count data from a financial system.

A New First-Order Integer-Valued Autoregressive Model with Bell Innovations

A Poisson distribution is commonly used as the innovation distribution for integer-valued autoregressive models, but its mean is equal to its variance, which limits flexibility, so a flexible, one-parameter, infinitely divisible Bell distribution may be a good alternative. In addition, for a parameter with a small value, the Bell distribution approaches the Poisson distribution. In this paper, we introduce a new first-order, non-negative, integer-valued autoregressive model with Bell innovations based on the binomial thinning operator. Compared with other models, the new model is not only simple but also particularly suitable for time series of counts exhibiting overdispersion. Some properties of the model are established here, such as the mean, variance, joint distribution functions, and multi-step-ahead conditional measures. Conditional least squares, Yule-Walker, and conditional maximum likelihood are used for estimating the parameters. Some simulation results are presented to access these estimates' performances. Real data examples are provided.

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.