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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. [ABSTRACT FROM AUTHOR]

Summary: Data related to the counting of elements of variable character are frequently encountered in time series studies. This paper brings forward a new class of k$$ k $$th‐order dependence‐driven random coefficient mixed thinning integer‐valued autoregressive time series model (DDRCMTINAR(k$$ k $$)) to deal with such data. Stationarity and ergodicity properties of the proposed model are derived in detail. The unknown parameters are estimated by conditional least squares, and modified quasi‐likelihood and asymptotic normality of the obtained parameter estimators is established. The performances of the adopted estimate methods are checked via simulations, which present that modified quasi‐likelihood estimators perform better than the conditional least squares considering the proportion of within‐Ω$$ \Omega $$ estimates in certain regions of the parameter space. The validity and practical utility of the model are investigated by epileptic seizure data and COVID‐19 data of suspected cases in China. [ABSTRACT FROM AUTHOR]

In this paper, a time-varying first-order mixture integer-valued threshold autoregressive process driven by explanatory variables is introduced. The basic probabilistic and statistical properties of this model are studied in depth. We proceed to derive estimators using the conditional least squares (CLS) and conditional maximum likelihood (CML) methods, while also establishing the asymptotic properties of the CLS estimator. Furthermore, we employed the CLS and CML score functions to infer the threshold parameter. Additionally, three test statistics to detect the existence of the piecewise structure and explanatory variables were utilized. To support our findings, we conducted simulation studies and applied our model to two applications concerning the daily stock trading volumes of VOW. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new bivariate random coefficient autoregressive (BOD-RCAR(1)) process driven by both explanatory variables and past observations. Firstly, some statistical properties of this model are derived. Secondly, three methods are used for estimating the unknown parameters: conditional least squares (CLS), conditional maximum likelihood (CML) and maximum empirical likelihood (MEL). The asymptotic properties of the estimators are given. Besides, two kinds of test based on empirical likelihood (EL) are established. A simulation experiment is presented to demonstrate the performance of the proposed method. Finally, an application to a real data example is investigated to assess the performance of the model. [ABSTRACT FROM AUTHOR]

To capture the bivariate count time series showing piecewise phenomena, we introduce a first-order bivariate threshold Poisson integer-valued autoregressive process. Basic probabilistic and statistical properties of the model are discussed. Conditional least squares and conditional maximum likelihood estimators, as well as their asymptotic properties, are obtained for both the cases that the threshold parameter is known or not. A new algorithm to estimate the threshold parameter of the model is also provided. Moreover, the nonlinearity test and forecasting problems are also addressed. Finally, some numerical results of the estimates and a real data example are presented. [ABSTRACT FROM AUTHOR]

In this article, a new autoregressive process for finite-range time series of counts is proposed to analyse the finite-range integer-valued data based on an invisible Markov chain. We derive the probabilistic and statistical properties of the model. Conditional least squares (CLS) method and conditional maximum likelihood (CML) method are employed to estimate the parameters of interest. Furthermore, the forecasting problem is addressed. In addition, multiple simulation studies are performed to investigate the finite sample performance of parameter estimators and to compare the proposed estimation methods. The proposed model is applied to a finite-range data series of measles infections in Germany in 2004–2005. [ABSTRACT FROM AUTHOR]

This paper uses the empirical likelihood (EL) method for a new random coefficient autoregressive process driven by explanatory variables and past observations through logistic structure (OD-RCAR (1)), which combines explanatory variables and past observations, and puts forward the penalized maximum empirical likelihood (PMEL) method for parameters estimation and variable selection. Firstly, limiting distributions of the estimating function and log empirical likelihood ratio statistics based on EL are established. Meanwhile, this paper sets up a confidence region and EL test for parameters. Secondly, the maximum empirical likelihood estimators and their asymptotic properties are obtained. At the same time, the penalized empirical likelihood ratio test statistic is given. Thirdly, it is proved in a high-dimensional setting that the PMEL in our model can solve the problem of order selection and parameter estimation. Finally, not only practical data applications but also numerical simulations are adopted in order to describe the performance of proposed methods. [ABSTRACT FROM AUTHOR]

This paper studies the empirical likelihood method for a multivariate first-order random coefficient autoregressive (RCAR(1)) model. Based on the modified least square score equation, empirical likelihood method is used to test the randomness of coefficients and estimate the unknown parameters. First, the limiting distribution of the log empirical likelihood ratio statistic is established. Second, the maximum empirical likelihood estimator is derived and its asymptotic properties are established. Finally, the performances of the estimator are compared with the modified least square estimator via simulation. Furthermore, simulation result shows that the test has the correct size and very good power for all cases considered. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new first-order mixture integer-valued threshold autoregressive process, based on the binomial and negative binomial thinning operators. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares (CLS) and conditional maximum likelihood (CML) estimators are derived and the asymptotic properties of the estimators are established. The inference for the threshold parameter is obtained based on the CLS and CML score functions. Moreover, the Wald test is applied to detect the existence of the piecewise structure. Simulation studies are considered, along with an application: the number of criminal mischief incidents in the Pittsburgh dataset • First-Order mixture integer-valued threshold autoregressive process based on binomial thinning and negative binomial thinning. • Parameter estimation for BiNB-MTINAR(1) process. • Detect the existence of the piecewise structure. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new first-order generalized Poisson integer-valued autoregressive process, for modeling integer-valued time series exhibiting a piecewise structure and overdispersion. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares and conditional maximum likelihood estimators are derived. The asymptotic properties of the estimators are established. Moreover, two special cases of the process are discussed. Finally, some numerical results of the estimates and a real data example are presented. [ABSTRACT FROM AUTHOR]

To analyse count time series data inflated at the r + 1 values { 0 , 1 , ... , r } , 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. [ABSTRACT FROM AUTHOR]

To better describe the characteristics of time series of counts such as overdispersion or structural change, in this paper, we redefines the integer-valued threshold autoregressive models based on negative binomial thinning (NBTINAR(1)) under a weaker condition that the expectation of the innovations is finite. Parameters' point estimation and interval estimation problems are considered. A method to test the nonlinearity of the data is provided. As an illustration, we conduct a simulation study and empirical analysis of Pittsburgh crime data sets. [ABSTRACT FROM AUTHOR]

In this paper, we study the use of the mean empirical likelihood (MEL) method in a first-order random coefficient integer-valued autoregressive model. The MEL ratio statistic is established, its limiting properties are discussed, and the confidence regions for the parameter of interest are derived. Furthermore, a simulation study is presented to demonstrate the performance of the proposed method. Finally, a real data analysis of dengue fever is performed. [ABSTRACT FROM AUTHOR]

In this article, we propose a new seasonal geometric integer-valued autoregressive process based on the negative binomial thinning operator with seasonal period s. Some basic probabilistic and statistical properties of the model are discussed. Conditional maximum likelihood estimators are obtained, and the asymptotic properties of the estimators are established. Some theoretical results of point forecasts are obtained. Numerical results are presented. At the end, two real data examples are investigated to assess the performance of our new model. [ABSTRACT FROM AUTHOR]

In this paper, a multinomial autoregressive model for finite-range time series of counts is introduced to analyze the finite-range integer-valued data with more than two states. Basic probabilistic and statistical properties of the model are discussed. The related estimations of the parameters in proposed model are considered using conditional least squares (CLS), weighted conditional least squares (WCLS) and conditional maximum likelihood (CML) methods. The asymptotic properties of the estimators are established. Some simulation studies are conducted to verify the proposed procedure. A real example is analyzed to illustrate the advantages of our model. • A multinomial autoregressive model for finite-range time series of counts. • Basic probabilistic and statistical properties. • The CLS, WCLS and CML estimations and the related asymptotic properties. • A real data example is investigated to assess the performance. [ABSTRACT FROM AUTHOR]

To better describe the data whose variance is greater than mean in time series analysis, this paper introduces the RCINAR(1) process with generalized negative binomial marginals. The related estimations of this process are considered using Yule-Walker, modified conditional least squares, conditional maximum likelihood and Bayesian methods. The asymptotic properties of the estimators are established. Some simulations are conducted to verify the proposed estimation methods and a real example is proposed to illustrate the advantages of our model. [ABSTRACT FROM AUTHOR]

In this paper, we consider the penalized estimation procedure for Poisson autoregressive model with sparse parameter structure. We study the theoretical properties of penalized conditional maximum likelihood (PCML) with several different penalties. We show that the penalized estimators perform as well as the true model was known. We establish the oracle properties of PCML estimators. Some simulation studies are conducted to verify the proposed procedure. A real data example is also provided. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new bivariate first-order random coefficient integer-valued autoregressive (BRCINAR(1)) process with dependent innovations. Some basic probabilistic and statistical properties of this model are obtained. Estimators of unknown parameters are derived by using Yule–Walker, conditional least squares and conditional maximum likelihood methods. The asymptotic properties of the estimators are established. The performance of these estimators is compared through a simulation experiment. Moreover, the coherent forecasting for BRCINAR(1) model is addressed. Finally, an application to a real data example is investigated to assess the performance of the model. • A new bivariate random coefficient INAR(1) (BRCINAR(1)) process is proposed. • Some basic probabilistic and statistical properties of this process are obtained. • Three different estimation methods are provided to estimate parameters. • The coherent forecasting for BRCINAR(1) process is addressed. • Coefficients follow BETA distribution and Gaussian mixture distribution are studied. • An application to a real data example is investigated. [ABSTRACT FROM AUTHOR]

The process of serially dependent counts with deflation or inflation of zeros is commonly observed in many applications. This paper investigates the monitoring of such a process, the first-order zero-modified geometric integer-valued autoregressive process (ZMGINAR(1)). In particular, two control charts, the upper-sided and lower-sided CUSUM charts, are developed to detect the shifts in the mean process of the ZMGINAR(1). Both the average run length performance and the standard deviation of the run length performance of these two charts are investigated by using Markov chain approaches. Also, an extensive simulation is conducted to assess the effectiveness or performance of the charts, and the presented methods are applied to two sets of real data arising from a study on the drug use. [ABSTRACT FROM AUTHOR]

To accurately and flexibly capture the dispersion features of time series of counts, we introduce the generalized Poisson thinning operation and further define some new integer-valued autoregressive processes. Basic probabilistic and statistical properties of the models are discussed. Conditional least squares and maximum quasi likelihood estimators are investigated via the moment targeting estimation methods for the innovation free case. Also, the asymptotic properties of the estimators are obtained. Conditional maximum likelihood estimation for the parametric cases are also discussed. Finally, some numerical results of the estimates and two real data examples are presented. [ABSTRACT FROM AUTHOR]

In this article, we study the problem of testing the constancy of the coefficient in a class of stationary first-order random coefficient autoregressive (RCAR(1)) model. We construct a new test statistic based on the locally most powerful-type (LMP) test. Under the null hypothesis, we derive the limiting distribution of the proposed test statistic. In the simulation, we compare the power between LMP test and empirical likelihood (EL) test and find that the accuracy of using LMP is 6.7%, 28.8%, and 26.1% higher than that of EL test under normal, student's t, and symmetric contamination errors, respectively. A real life data is given to illustrate the practical effectiveness of our test. [ABSTRACT FROM AUTHOR]

This article redefines the self-exciting threshold integer-valued autoregressive (SETINAR(2,1)) processes under a weaker condition that the second moment is finite, and studies the quasi-likelihood inference for the new model. The ergodicity of the new processes is discussed. Quasi-likelihood estimators for the model parameters and the asymptotic properties are obtained. Confidence regions of the parameters based on the quasi-likelihood method are given. A simulation study is conducted for the evaluation of the proposed approach and an application to a real data example is provided. [ABSTRACT FROM AUTHOR]

In this article, we develop a Bayesian analysis in autoregressive model with explanatory variables. When σ2is known, we consider a normal prior and give the Bayesian estimator for the regression coefficients of the model. For the case σ2is unknown, another Bayesian estimator is given for all unknown parameters under a conjugate prior. Bayesian model selection problem is also being considered under the double-exponential priors. By the convergence of ρ-mixing sequence, the consistency and asymptotic normality of the Bayesian estimators of the regression coefficients are proved. Simulation results indicate that our Bayesian estimators are not strongly dependent on the priors, and are robust. [ABSTRACT FROM PUBLISHER]

In this article, the problem of interest is testing the conditional heteroscedasticity of Poisson autoregressive model. We construct a non parametric test statistic based on empirical likelihood method. The asymptotic distribution of the proposed statistic is derived and its finite-sample property is examined through Monte Carlo simulations. The simulation results show that the proposed method is good for practical use. [ABSTRACT FROM PUBLISHER]

Multiply robust estimation with missing data is considered as an important field in statistics, which incorporates information by weighting multiply candidate models and loosens the requirement of the model specification. Nevertheless, in high-dimensional cases one more flexible hypothesis is the "true structure" beyond the correct model. In this paper, we study the parametric estimation for high-order autoregressive processes with a lagged-dependent binary explanatory variable that is missing at random (MAR). Based on the "true structure" specification, we propose a penalized multiply robust estimation equation in the presence of multiply candidate model sets. The selecting criterion for optimal tuning parameters is modified for the model identification with incomplete data. We validate that our tuning criterion can correctly distinguish the true autoregressive coefficients from zero asymptotically, the estimators of population parameters enjoy the oracle properties as well. Some simulations are carried out and we apply the method to fit the model for the U.S. Industrial Production Index data and produce out-of-sample forecasts to confirm the rationality of results. [ABSTRACT FROM AUTHOR]

In view of the complexity and asymmetry of finite range multi-state integer-valued time series data, we propose a first-order random coefficient multinomial autoregressive model in this paper. Basic probabilistic and statistical properties of the model are discussed. Conditional least squares (CLS) and weighted conditional least squares (WCLS) estimators of the model parameters are derived, and their asymptotic properties are established. In simulation studies, we compare these two methods with the conditional maximum likelihood (CML) method to verify the proposed procedure. A real example is applied to illustrate the advantages of our model. [ABSTRACT FROM AUTHOR]

This paper considers the periodic self-exciting threshold integer-valued autoregressive processes under a weaker condition in which the second moment is finite instead of the innovation distribution being given. The basic statistical properties of the model are discussed, the quasi-likelihood inference of the parameters is investigated, and the asymptotic behaviors of the estimators are obtained. Threshold estimates based on quasi-likelihood and least squares methods are given. Simulation studies evidence that the quasi-likelihood methods perform well with realistic sample sizes and may be superior to least squares and maximum likelihood methods. The practical application of the processes is illustrated by a time series dataset concerning the monthly counts of claimants collecting short-term disability benefits from the Workers' Compensation Board (WCB). In addition, the forecasting problem of this dataset is addressed. [ABSTRACT FROM AUTHOR]

To better describe the characteristics of time series of counts such as over-dispersion, asymmetry and structural change, this paper considers a class of integer-valued self-exciting threshold autoregressive processes that properly capture flexible asymmetric and nonlinear responses without assuming the distributions for the errors. Empirical likelihood methods are proposed for constructing confidence intervals for the parameters of interest. Maximum empirical likelihood estimators, as well as their asymptotic properties, are obtained for both the cases that the threshold variable is known or not. A method to test the nonlinearity of the data is provided. As an illustration, we conduct a simulation study and empirical analysis of Pittsburgh crime data sets. [ABSTRACT FROM AUTHOR]

The paper develops a first-order random coefficient mixed-thinning integer-valued autoregressive time series model (RCMTINAR(1)) to deal with the data related to the counting of elements of variable character. Moments and autocovariance functions for this model are studied as the distribution of the innovation sequence is unknown. The conditional least squares and modified quasi-likelihood are adopted to estimate the model parameters. Asymptotic properties of the obtained estimators are established. The performances of these estimators are investigated and compared with false modified quasi-likelihood via simulations. Finally, the practical relevance of the model is illustrated by using two applications to a SIMpass data set and a burglary data set with a comparison with relevant models that exist so far in the literature. [ABSTRACT FROM AUTHOR]