Integer-valued Autoregressive Model Research Articles

A class of mixed thinning threshold integer-valued autoregressive model for the COVID-19 data

To better capture the features of time series of counts such as the counting of elements of variable character and piecewise phenomenon, this paper proposes a class of mixed-thinning threshold integer-valued autoregressive model where the distribution of the innovation sequence is unknown. Strict stationarity ergodicity property is proved. The unknown parameters in the model are estimated by conditional least squares and modified quasi-likelihood methods, as well as their asymptotic normality, are established for the case that the threshold variable is known. The performances of the estimators are investigated in simulation, which manifests that the modified quasi-likelihood estimation performs better. Finally, a set of COVID-19 data on critical cases imported from outside China is analyzed to illustrate the application of developed model.

Change point detection in INAR(p) models via likelihood ratio scanning method

In this paper, we consider an integer-valued autoregressive (INAR) model. Based on the likelihood ratio scanning method (LRSM), the numbers and locations of the piecewise stationary INAR process change points are discussed. The minimum description length (MDL) function of the INAR model is given. Moreover, we also study the case when the sum of coefficients in each segment of the model tends to 1 and make adjustments to the MDL criterion in the LRSM. In this way, the accuracy of change point detection is improved. In the end, simulation experiments and real data analysis are conducted to illustrate the efficiency of the LRSM.

Maximum-likelihood estimation of the Po-MDDRCINAR(p) model with analysis of a COVID-19 data

Integer-valued data are frequently encountered in time series studies. A pth-order mixed dependence-driven random coefficient integer-valued autoregressive time series model (Po-MDDRCINAR(p)) in view of binomial and negative binomial operators, where the innovation sequence follows a Poisson distribution, is investigated to provide meaningful theoretical explanations. Strict stationary and ergodicity of the model are demonstrated. Furthermore, the conditional least-squares and conditional maximum-likelihood methods are adopted to estimate the parameters, where the asymptotic characterization of the estimators is derived. Finite-sample properties of the conditional maximum-likelihood estimator are examined in relation to the widely used conditional least-squares estimator. The conclusion is that, if the Poisson assumption of the innovation sequence can be justified, conditional maximum-likelihood method performs better in terms of MADE and MSE. Finally, the practical performance of the model is illustrated by a set of COVID-19 data of suspected cases in China with a comparison with relevant models that exist so far in the literature.

A non-linear integer-valued autoregressive model with zero-inflated data series

A new non-linear stationary process for time series of counts is introduced. The process is composed of the survival and innovation component. The survival component is based on the generalized zero-modified geometric thinning operator, where the innovation process figures in the survival component as well. A few probability distributions for the innovation process have been discussed, in order to adjust the model for observed series with the excess number of zeros. The conditional maximum likelihood and the conditional least squares methods are investigated for the estimation of the model parameters. The practical aspect of the model is presented on some real-life data sets, where we observe data with inflation as well as deflation of zeroes so we can notice how the model can be adjusted with the proper parameter selection.

Modelling ℤ-valued time series with Skellam thinning-based INAR(1) process

ABSTRACT Non-stationary count time series, which are small in value and show a trend having relatively large fluctuation, are commonly encountered in real-world applications. Differencing is a commonly employed method to transform a non-stationary time series into a stationary one. However, the differenced count time series is defined on Z and cannot be fitted by the mainstream count time series models, which are defined on N 0 . Although some integer-valued autoregressive (INAR) models based on the signed thinning operator have been proposed for fitting Z -valued time series, these models still encounter difficulties in terms of data generation mechanism, model selection, and parameter estimation. These difficulties may pose challenges for practitioners in practical applications. Therefore, this article utilizes the reparameterized version of the Skellam (Poisson difference) distribution to construct a new thinning operator, thereby introducing a class of Z -valued time series models. Strict stationarity, ergodicity, statistical properties, and parameter estimation methods of the processes are detailedly discussed. An application to the monthly counts of newly listed companies illustrates that the new model is superior in data analysis of finance and economics.

Integer-valued autoregressive models based on quasi Pólya thinning operator

Integer-valued autoregressive models based on quasi Pólya thinning operator

Threshold integer-valued autoregressive model with serially dependent innovation

ABSTRACT Nonlinearity in count time series is commonly encountered in practice. To better explain the nonlinear phenomena in count time series, this article introduces a new threshold INAR(1) model using the idea of serially dependent innovation. The proposed model contains some existing INAR(1) models as special cases and establishes a connection between the threshold INAR(1) model and the threshold INARCH(1) model. The basic probabilistic and statistical properties of the new model are investigated. Model parameters, including the threshold variable, are estimated by the conditional least squares, modified quasi-likelihood and conditional maximum likelihood methods. Asymptotic properties and numerical results of the estimates are also studied. An application to the monthly trading volume of stock B in Shanghai Stock Exchange is conducted to show the practicability of the proposed model.

Goodness-of-fit testing in bivariate count time series based on a bivariate dispersion index

AbstractA common choice for the marginal distribution of a bivariate count time series is the bivariate Poisson distribution. In practice, however, when the count data exhibit zero inflation, overdispersion or non-stationarity features, such that a marginal bivariate Poisson distribution is not suitable. To test the discrepancy between the actual count data and the bivariate Poisson distribution, we propose a new goodness-of-fit test based on a bivariate dispersion index. The asymptotic distribution of the test statistic under the null hypothesis of a first-order bivariate integer-valued autoregressive model with marginal bivariate Poisson distribution is derived, and the finite-sample performance of the goodness-of-fit test is analyzed by simulations. A real-data example illustrate the application and usefulness of the test in practice.

A zero-inflated Poisson integer-valued autoregressive model with time-varying coefficients covariates

For integer-valued time series, we may be interested in how a covariate affects the series. In the existing models, it is commonly assumed that the coefficients of the covariates are constant, and the innovation process follows a Poisson distribution. However, in practice, the influence of covariates may vary with time, and there may be a large number of zeros. To address this situation, this paper proposes an integer-valued autoregressive model with time-varying coefficients on covariates, incorporating a zero-inflated Poisson innovation process. The binomial thinning parameters and unknown functions of the proposed model are estimated based on approximate conditional least squares and the cubic spline method. The consistency and asymptotic properties of the binomial thinning parameters are investigated. Simulation studies and real data analysis support the flexibility and good performance of the proposed model.

A note on the asymptotic behavior of a mildly unstable integer-valued AR(1) model

This note examines the conditional least squares (CLS) estimators for integer-valued autoregressive (INAR) models, with a focus on the INAR(1) model. Our investigation reveals that the joint limiting distribution of the CLS estimators for the autoregressive coefficient and the mean of the model errors is degenerate in the case of mildly unstable INAR(1) models with moderate deviations from unity. This degeneracy may pose challenges for statistical inference. We provide commenting notes based on the asymptotic results to address this issue. Additionally, we conduct simulation studies and analyze a real-world dataset to validate the theoretical findings.

Statistical Models for Outbreak Detection of Measles in North Cotabato, Philippines

A measles outbreak occurs when the number of cases of measles in the population exceeds the typical level. Outbreaks that are not detected and managed early can increase mortality and morbidity and incur costs from activities responding to these events. The number of measles cases in the Province of North Cotabato, Philippines, was used in this study. Weekly reported cases of measles from January 2016 to December 2021 were provided by the Epidemiology and Surveillance Unit of the North Cotabato Provincial Health Office. Several integer-valued autoregressive (INAR) time series models were used to explore the possibility of detecting and identifying measles outbreaks in the province along with the classical ARIMA model. These models were evaluated based on goodness of fit, measles outbreak detection accuracy, and timeliness. The results of this study confirmed that INAR models have the conceptual advantage over ARIMA since the latter produces non-integer forecasts, which are not realistic for count data such as measles cases. Among the INAR models, the ZINGINAR (1) model was recommended for having a good model fit and timely and accurate detection of outbreaks. Furthermore, policymakers and decision-makers from relevant government agencies can use the ZINGINAR (1) model to improve disease surveillance and implement preventive measures against contagious diseases beforehand.

Negative binomial community network vector autoregression for multivariate integer-valued time series

Modeling multivariate integer-valued time series with appropriate methods is currently a popular research topic. In this paper, we propose a multivariate integer-valued autoregressive time series model based on a fixed network community structure. We use the negative binomial distribution as the conditional marginal distribution and a copula to construct the conditional joint distribution. The newly proposed model introduces the heterogeneity of nodes. Stability conditions are provided for both fixed and increasing dimensions. We estimate the parameters of the proposed model by maximizing the quasi-likelihood function with known and unknown community membership matrices, respectively. Corresponding asymptotic properties of parameter estimates are also provided. A simulation study is conducted to demonstrate the asymptotic behavior of the proposed model, and two real datasets are employed to compare the proposed model with other competitive models.

First-order multivariate integer-valued autoregressive model with multivariate mixture distributions

The univariate integer-valued time series has been extensively studied, but literature on multivariate integer-valued time series models is quite limited and the complex correlation structure among the multivariate integer-valued time series is barely discussed. In this study, we proposed a first-order multivariate integer-valued autoregressive model to characterize the correlation among multivariate integer-valued time series with higher flexibility. Under the general conditions, we established the stationarity and ergodicity of the proposed model. With the proposed method, we discussed the models with multivariate Poisson-lognormal distribution and multivariate geometric-logitnormal distribution and the corresponding properties. The estimation method based on EM algorithm was developed for the model parameters and extensive simulation studies were performed to evaluate the effectiveness of proposed estimation method. Finally, a real crime dataset was analysed to demonstrate the advantage of the proposed model with comparison to the other models.

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.

A MIXTURE INTEGER-VALUED AUTOREGRESSIVE MODEL WITH A STRUCTURAL BREAK

In this manuscript we introduce a mixture integer-valued autoregressive model with a structural break. The introduced model is a mixture of an INAR(1) model with the binomial thinning operator and an INAR(1) model with the negative binomial thinning operator. Some properties of the introduced model are derived. The unknown parameters of the model are estimated by some methods and the performances of the obtained estimators are checked by simulations. At the end of the paper, two possible applications of the model are provided and discussed.

On Comparing and Assessing Robustness of Some Popular Non-Stationary BINAR(1) Models

Intra-day transactions of stocks from competing firms in the financial markets are known to exhibit significant volatility and over-dispersion. This paper proposes some bivariate integer-valued auto-regressive models of order 1 (BINAR(1)) that are useful to analyze such financial series. These models were constructed under both time-variant and time-invariant conditions to capture features such as over-dispersion and non-stationarity in time series of counts. However, the quest for the most robust BINAR(1) models is still on. This paper considers specifically the family of BINAR(1)s with a non-diagonal cross-correlation structure and with unpaired innovation series. These assumptions relax the number of parameters to be estimated. Simulation experiments are performed to assess both the consistency of the estimators and the robust behavior of the BINAR(1)s under mis-specified innovation distribution specifications. The proposed BINAR(1)s are applied to analyze the intra-day transaction series of AstraZeneca and Ericsson. Diagnostic measures such as the root mean square errors (RMSEs) and Akaike information criteria (AICs) are also considered. The paper concludes that the BINAR(1)s with negative binomial and COM–Poisson innovations are among the most suitable models to analyze over-dispersed intra-day transaction series of stocks.

The Circumstance-Driven Bivariate Integer-Valued Autoregressive Model.

The novel circumstance-driven bivariate integer-valued autoregressive (CuBINAR) model for non-stationary count time series is proposed. The non-stationarity of the bivariate count process is defined by a joint categorical sequence, which expresses the current state of the process. Additional cross-dependence can be generated via cross-dependent innovations. The model can also be equipped with a marginal bivariate Poisson distribution to make it suitable for low-count time series. Important stochastic properties of the new model are derived. The Yule-Walker and conditional maximum likelihood method are adopted to estimate the unknown parameters. The consistency of these estimators is established, and their finite-sample performance is investigated by a simulation study. The scope and application of the model are illustrated by a real-world data example on sales counts, where a soap product in different stores with a common circumstance factor is investigated.

Comparison of estimation and prediction methods for a zero-inflated geometric INAR(1) process with random coefficients

This study explores zero-inflated count time series models used to analyze data sets with characteristics such as overdispersion, excess zeros, and autocorrelation. Specifically, we investigate the ZIGINAR RC ( 1 ) process, a first-order stationary integer-valued autoregressive model with random coefficients and a zero-inflated geometric marginal distribution. Our focus is on examining various estimation and prediction techniques for this model. We employ estimation methods, including Whittle, Taper Spectral Whittle, Maximum Empirical Likelihood, and Sieve Bootstrap estimators for parameter estimation. Additionally, we propose forecasting approaches, such as median, Bayesian, and Sieve Bootstrap methods, to predict future values of the series. We assess the performance of these methods through simulation studies and real-world data analysis, finding that all methods perform well, providing 95% highest predicted probability intervals that encompass the observed data. While Bayesian and Bootstrap methods require more time for execution, their superior predictive accuracy justifies their use in forecasting.

A Two-Step Estimation Method for a Time-Varying INAR Model

This paper proposes a new time-varying integer-valued autoregressive (TV-INAR) model with a state vector following a logistic regression structure. Since the autoregressive coefficient in the model is time-dependent, the Kalman-smoothed method is applicable. Some statistical properties of the model are established. To estimate the parameters of the model, a two-step estimation method is proposed. In the first step, the Kalman-smoothed estimation method, which is suitable for handling time-dependent systems and nonstationary stochastic processes, is utilized to estimate the time-varying parameters. In the second step, conditional least squares is used to estimate the parameter in the error term. This proposed method allows estimating the parameters in the nonlinear model and deriving the analytical solutions. The performance of the estimation method is evaluated through simulation studies. The model is then validated using actual time series data.

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.