Integer-valued Autoregressive Research Articles

Bivariate Random Coefficient Integer-Valued Autoregressive Model Based on a ρ-Thinning Operator

While overdispersion is a common phenomenon in univariate count time series data, its exploration within bivariate contexts remains limited. To fill this gap, we propose a bivariate integer-valued autoregressive model. The model leverages a modified binomial thinning operator with a dispersion parameter ρ and integrates random coefficients. This approach combines characteristics from both binomial and negative binomial thinning operators, thereby offering a flexible framework capable of generating counting series exhibiting equidispersion, overdispersion, or underdispersion. Notably, our model includes two distinct classes of first-order bivariate geometric integer-valued autoregressive models: one class employs binomial thinning (BVGINAR(1)), and the other adopts negative binomial thinning (BVNGINAR(1)). We establish the stationarity and ergodicity of the model and estimate its parameters using a combination of the Yule–Walker (YW) and conditional maximum likelihood (CML) methods. Furthermore, Monte Carlo simulation experiments are conducted to evaluate the finite sample performances of the proposed estimators across various parameter configurations, and the Anderson-Darling (AD) test is employed to assess the asymptotic normality of the estimators under large sample sizes. Ultimately, we highlight the practical applicability of the examined model by analyzing two real-world datasets on crime counts in New South Wales (NSW) and comparing its performance with other popular overdispersed BINAR(1) models.

EXPLORING THE INAR MODEL ON HEAVY TAILED TIME SERIES DATA WITH OUTLIERS

Count data are intrinsically measures of event frequency; it is clear that there is an intrinsic relationship with recurring time to event. Events are typically tallied within time intervals for practical and convenient reasons. The existence of outliers is one issue that prevents count data from being stationary in time series analysis; this has an impact on the effectiveness of fitting several common stationary models to the count data collected over time. Thus, the purpose of this study was to examine how well the Integer Valued Autoregressive (INAR) model performed while modeling count data that included outliers. While this model has been studied for count time series data, it has not been studied for varying degrees of outliers. A monte-carlo simulation was carried out to select the best INAR(p), where p=1,2,3 and 4 on data with 10%, 20% and 30% outliers at different sample sizes. The INAR (4) has the best fit across the sample sizes at the larger percentages of outliers while INAR (3) at the lowest percentage with smallest information criteria of assessment and they are therefore recommended for such modeling.

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.

New One-Parameter Over-Dispersed Discrete Distribution and Its Application to the Nonnegative Integer-Valued Autoregressive Model of Order One

Count data arise in inference, modeling, prediction, anomaly detection, monitoring, resource allocation, evaluation, and performance measurement. This paper focuses on a one-parameter discrete distribution obtained by compounding the Poisson and new X-Lindley distributions. The probability-generating function, moments, skewness, kurtosis, and other properties are derived in the closed form. The maximum likelihood method, method of moments, least squares method, and weighted least squares method are used for parameter estimation. A simulation study is carried out. The proposed distribution is applied as the innovation in an INAR(1) process. The importance of the proposed model is confirmed through the analysis of two real datasets.

High-Order Self-excited Threshold Integer-Valued Autoregressive Model: Estimation and Testing

High-Order Self-excited Threshold Integer-Valued Autoregressive Model: Estimation and Testing

On Periodic Generalized Poisson INAR(1) Model

In this paper, we introduce a first-order Periodic Generalized Poisson Integer-Valued Autoregressive model which has been shown to be useful to describe overdispersion, equidispersion and underdispersion feature encountered in periodically correlated Integer-Valued time series. Some probabilistic and statistical properties are established, such as the periodically correlated stationarity conditions, in the first and the second moments are provided and the closed-forms of these moments are, under these conditions, derived. Moreover, the structure of the periodic autocovariance is obtained. The estimation problem is addressed through the Yule-Walker the Two-Stage Conditional Least Squares and the Conditional Maximum Likelihood methods. The performance of these methods is done through an intensive simulation study and an application on real data set is accomplished. Keywords and phrases: Periodic Generalized Poisson, Integer-Valued Autoregressive, Periodically correlated process, periodically stationary condition.

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.

Integer-valued Autoregressive Model Based on Innovations with Discrete Exponential-Weibull Distribution

Integer-valued Autoregressive Model Based on Innovations with Discrete Exponential-Weibull Distribution

Non-linear INAR(1) processes under an alternative geometric thinning operator

We propose a novel class of first-order integer-valued AutoRegressive (INAR(1)) models based on a new operator, the so-called geometric thinning operator, which induces a certain non-linearity to the models. We show that this non-linearity can produce better results in terms of prediction when compared to the linear case commonly considered in the literature. The new models are named non-linear INAR(1) (in short NonLINAR(1)) processes. We explore both stationary and non-stationary versions of the NonLINAR processes. Inference on the model parameters is addressed and the finite-sample behavior of the estimators investigated through Monte Carlo simulations. Two real data sets are analyzed to illustrate the stationary and non-stationary cases and the gain of the non-linearity induced for our method over the existing linear methods. A generalization of the geometric thinning operator and an associated NonLINAR process are also proposed and motivated for dealing with zero-inflated or zero-deflated count time series data.

Asymptotic behaviour of the portmanteau tests in an integer-valued AR model

The portmanteau test has been popular for diagnostic checking in time series models. Asymptotic properties of portmanteau tests have been exhaustively studied for real-valued time series model though, similar results for integer-valued autoregressive (INAR) models are not well documented, nevertheless. In view of this, we investigate the asymptotic behaviour of the Box-Pierce and Ljung-Box portmanteau tests in an INAR model. It turns out that these tests are chi-squared distributed asymptotically under mild conditions regardless of the process being stable or nearly unstable.

Semiparametric estimation of INAR models using roughness penalization

Popular models for time series of count data are integer-valued autoregressive (INAR) models, for which the literature mainly deals with parametric estimation. In this regard, a semiparametric estimation approach is a remarkable exception which allows for estimation of the INAR models without any parametric assumption on the innovation distribution. However, for small sample sizes, the estimation performance of this semiparametric estimation approach may be inferior. Therefore, to improve the estimation accuracy, we propose a penalized version of the semiparametric estimation approach, which exploits the fact that the innovation distribution is often considered to be smooth, i.e. two consecutive entries of the PMF differ only slightly from each other. This is the case, for example, in the frequently used INAR models with Poisson, negative binomially or geometrically distributed innovations. For the data-driven selection of the penalization parameter, we propose two algorithms and evaluate their performance. In Monte Carlo simulations, we illustrate the superiority of the proposed penalized estimation approach and argue that a combination of penalized and unpenalized estimation approaches results in overall best INAR model fits.

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.

Improving the Estimation and Predictions of Small Time Series Models

Abstract A new approach is developed for improving the point estimation and predictions of parametric time-series models. The method targets performance criteria such as estimation bias, root mean squared error, variance, or prediction error, and produces closed-form estimators focused towards these targets via a computational approximation method. This is done for an autoregression coefficient, for the mean reversion parameter in Vasicek and CIR diffusion models, for the binomial thinning parameter in integer-valued autoregressive (INAR) models, and for predictions from a CIR model. The success of the prediction targeting approach is shown in Monte Carlo simulations and in out-of-sample forecasting of the US Federal Funds rate.

Cluster point processes and Poisson thinning INARMA

In this paper, we consider Poisson thinning Integer-valued time series models, namely integer-valued moving average model (INMA) and Integer-valued Autoregressive Moving Average model (INARMA), and their relationship with cluster point processes, the Cox point process and the dynamic contagion process. We derive the probability generating functionals of INARMA models and compare to that of cluster point processes. The main aim of this paper is to prove that, under a specific parametric setting, INMA and INARMA models are just discrete versions of continuous cluster point processes and hence converge weakly when the length of subintervals goes to zero.

Modeling digital camera monitoring count data with intermittent zeros for short-term prediction

Digital camera monitoring has revolutionised survey designs in many fields, as an important source of information. The extended sampling coverage offered by this monitoring scheme makes it preferable compared to other traditional methods of survey. However, data obtained from digital camera monitoring are often highly variable, and characterized by sparse periods of zero counts, interspersed with missing observations due to outages. In practice, missing data of relatively shorter duration are mostly observed and are often imputed using interpolation techniques, ignoring long-term trends leading to inherent estimation biases. In this study, we investigated time series forecasting methods that adequately handle intermittency and produced plausible estimates for imputation and forecasting purposes. The study utilised a yearlong digital camera monitoring data set of hourly counts of powerboat launches at three boat ramps in Western Australia. Several time series forecasting methods were evaluated and the accuracies of their point estimates of forecasts for various lead times in hours of up to one week were assessed using cross-validation techniques. Intermittent demand forecasting techniques, including Croston's method and Syntetos-Boylan Approximation (SBA) models, and count data forecasting methods including autoregressive conditional Poisson (ACP) models, integer-valued moving average (INMA) models, and integer-valued autoregressive (INAR) models were evaluated. ACP and INAR models performed better than intermittent demand forecasting techniques for short forecast horizons and provided some evidence of their sufficiency in predicting the dynamics in recreational boating activities. This result established that, in as much as intermittency may be a key feature for a given dataset, it should not override the systemic characteristics of data in the application of forecasting techniques. Our results provide plausible estimates for short-term missing data and forecasts for monitoring events, with applications in supporting proper tracking of usage of facilities, guiding resource allocations and providing insightful perspectives for management decisions.

A new INAR model based on Poisson-BE2 innovations

In this paper, two-parameter Poisson binomial-exponential 2 (PBE2) distribution is firstly reviewed, then a new integer-valued autoregressive (INAR) model with PBE2 innovations is proposed. The definition and statistical properties of the proposed model are given, including the mean, variance, covariance, strict stationarity and ergodicity. Two-step conditional least squares and conditional maximum likelihood estimation methods are considered to estimate the parameters. To assess the proposed model, a crime data set is analyzed and a comparison with other competing INAR models is given, which shows that the proposed model yields a better performance.

A class of max-INAR(1) processes with explanatory variables

The sample paths generated by the integer-valued autoregressive (INAR) processes usually fluctuate up and down in both directions. In some real examples, we will encounter that the path increases in one direction, then decreases, and then goes back and forth. If such count data sets are concerned, the conventional INAR processes cannot be able to capture this up-and-down movement tendency. To solve this problem, the max-AR process was extended to the discrete case, called max-INAR(1) process. To make this process more flexible, we introduce a class of max-INAR processes with explanatory variables, where the autoregressive parameter in the thinning operator depends on explanatory variables. The conditional maximum likelihood method is used to estimate parameters. Further, we consider the one-step forecast for the proposed models. The performance of the estimators in finite sample is numerically assessed. The proposed models are also applied to three real examples.

On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations

Abstract We investigate joint temporal and contemporaneous aggregation of N independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient α ∈ (0, 1) and with idiosyncratic Poisson innovations. Assuming that α has a density function of the form ψ(x) (1 − x) β , x ∈ (0, 1), with β ∈ (−1, ∞) and lim x ↑ 1 ψ ( x ) = ψ 1 ∈ ( 0 , ∞ ) $\lim\limits_{x\uparrow 1} \psi(x) = \psi_1 \in (0, \infty)$ , different limits of appropriately centered and scaled aggregated partial sums are shown to exist for β ∈ (−1, 0] in the so-called simultaneous case, i.e., when both N and the time scale n increase to infinity at a given rate. The case β ∈ (0, ∞) remains open. We also give a new explicit formula for the joint characteristic functions of finite dimensional distributions of the appropriately centered aggregated process in question.

Integer-valued autoregressive processes with prespecified marginal and innovation distributions: a novel perspective

Integer-valued autoregressive (INAR) processes are generally defined by specifying the thinning operator and either the innovations or the marginal distributions. The major limitations of such processes include difficulties in deriving the marginal properties and justifying the choice of the thinning operator. To overcome these drawbacks, we propose a novel approach for building an INAR model that offers the flexibility to prespecify both marginal and innovation distributions. Thus, the thinning operator is no longer subjectively selected but is rather a direct consequence of the marginal and innovation distributions specified by the modeler. Novel INAR processes are introduced following this perspective; these processes include a model with geometric marginal and innovation distributions (Geo-INAR) and models with bounded innovations. We explore the Geo-INAR model, which is a natural alternative to the classical Poisson INAR model. The Geo-INAR process has interesting stochastic properties, such as MA() representation, time reversibility, and closed forms for the -order transition probabilities, which enables a natural framework to perform coherent forecasting. To demonstrate the real-world application of the Geo-INAR model, we analyze a count time series of criminal records in sex offenses using the proposed methodology and compare it with existing INAR and integer-valued generalized autoregressive conditional heteroscedastic models.