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In this paper, we consider Bayesian methods for analyzing time series of count data under a Poisson regression model with a latent auto-regressive process embedded as an additive error term. We propose two different methods; the first method samples the latent variables one by one while the second method samples them jointly. The two methods are compared by simulation studies and an example employing real data. In terms of relative bias and root-mean-squared-errors, the two methods perform almost the same. However, the mixing performance of the first method is better than the second method for most of the simulation scenarios. [ABSTRACT FROM AUTHOR]

In this article, we consider a first-order integer-valued autoregressive (INAR(1)) model. Then, we propose change point estimators for the rate and dependence parameters in INAR(1) model using maximum likelihood estimation method when the type of change belongs to a family of monotonic changes. To monitor the process, a combined EWMA and c control chart is considered. The results show that the proposed change point estimators provide efficient estimates of the change time. At the end, to illustrate the application of the proposed estimators, a real case related to IP counts data is investigated. [ABSTRACT FROM AUTHOR]

In this article, we present the explicit expressions for the higher-order moments and cumulants of the first-order random coefficient integer-valued autoregressive (RCINAR(1)) process. The spectral and bispectral density functions are also obtained, which can characterize the RCINAR(1) process in the frequency domain. We use a frequency domain approach which is named Whittle criterion to estimate the parameters of the process. We propose a test statistic which is based on the frequency domain approach for the hypothesis test,H0: α = 0⟷H1: 0 < α < 1, where α is the mean of the random coefficient in the process. The asymptotic distribution of the test statistic is obtained. We compare the proposed test statistic with other statistics that can test serial dependence in time series of count via a typically numerical simulation, which indicates that our proposed test statistic has a good power. [ABSTRACT FROM AUTHOR]

To monitor count-type observations that are correlated over time, the Poisson INAR(1) CUSUM chart has been proposed for cases where an integer-valued autoregressive process of order 1 with a Poisson marginal is appropriate. In a recent study, it is shown that the Poisson INAR(1) CUSUM control chart is especially well suited to detection problems that involve mean shifts in a process. Although special causes of variation that result in mean shifts are the most common out-of-control situations in practice, some special causes of variation may also result in shifts of the autocorrelation coefficient or the process variance. In this research, we define three types of residuals from the time-series model and develop corresponding approaches for CUSUM monitoring. In particular, three CUSUM control charts, one for each of these residuals, as well as a multivariate CUSUM monitoring of the residuals are investigated. A Phase II comparison of these residuals' monitoring approaches together with a benchmark CUSUM of the count type observations indicate that, for different types of process shifts, different residual monitoring approaches may perform better. [ABSTRACT FROM AUTHOR]

Knowing the time of change would narrow the search to find and identify the variables disturbing a process. The knowledge of the change point can greatly aid practitioners in detecting and removing the special cause(s). Count processes with an autocorrelation structure are commonly observed in real-world applications and can often be modeled by the first-order integer-valued autoregressive (INAR) model. The most widely used marginal distribution for count processes is Poisson. In this study, change-point estimators are proposed for the parameters of correlated Poisson count processes. To do this, Newton's method is first used to approximate the parameters of the process. Then, maximum likelihood estimators of the process change point are developed. The performances of these estimators are next evaluated when they are employed in a combined exponentially weighted moving average (EWMA) and c scheme. Finally, for the rate parameter, the proposed estimator is compared with the estimator developed for independent observations. [ABSTRACT FROM AUTHOR]

In recent years, there has been a growing interest in modelling integred-valued time series. In this article, we propose a modified and generalized version of the first order rounded integer-valued autoregressive RINAR(1) model, originally introduced by Kachour and Yao (2009). Indeed, this class can be considered as an alternative of classical models based on the thinning operators. Using a Markov chain method, conditions for stationarity and the existence of moments are investigated. Least squares estimator of the model parameters is considered and its consistence is established. Finally, we describe the price change data using a model of the new class. [ABSTRACT FROM PUBLISHER]

The Poisson distribution is a simple and popular model for count-data random variables, but it suffers from the equidispersion requirement, which is often not met in practice. While models for overdispersed counts have been discussed intensively in the literature, the opposite phenomenon, underdispersion, has received only little attention, especially in a time series context. We start with a detailed survey of distribution models allowing for underdispersion, discuss their properties and highlight possible disadvantages. After having identified two model families with attractive properties as well as only two model parameters, we combine these models with the INAR(1) model (integer-valuedautoregressive), which is particularly well suited to obtain auotocorrelated counts with underdispersion. Properties of the resulting stationary INAR(1) models and approaches for parameter estimation are considered, as well as possible extensions to higher order autoregressions. Three real-data examples illustrate the application of the models in practice. [ABSTRACT FROM PUBLISHER]

The modelling and analysis of count-data time series are areas of emerging interest with various applications in practice. We consider the particular case of the binomial AR(1) model, which is well suited for describing binomial counts with a first-order autoregressive serial dependence structure. We derive explicit expressions for the joint (central) moments and cumulants up to order 4. Then, we apply these results for expressing moments and asymptotic distribution of the squared difference estimator as an alternative to the sample autocovariance. We also analyse the asymptotic distribution of the conditional least-squares estimators of the parameters of the binomial AR(1) model. The finite-sample performance of these estimators is investigated in a simulation study, and we apply them to real data about computerized workstations. [ABSTRACT FROM AUTHOR]

Multivariate count time series data occur in many different disciplines. The class of INteger-valued AutoRegressive (INAR) processes has the great advantage to consider explicitly both the discreteness and autocorrelation characterizing this type of data. Moreover, extensions of the simple INAR(1) model to the multi-dimensional space make it possible to model more than one series simultaneously. However, existing models do not offer great flexibility for dependence modelling, allowing only for positive correlation. In this work, we consider a bivariate INAR(1) (BINAR(1)) process where cross-correlation is introduced through the use of copulas for the specification of the joint distribution of the innovations. We mainly emphasize on the parametric case that arises under the assumption of Poisson marginals. Other marginal distributions are also considered. A short application on a bivariate financial count series illustrates the model. [ABSTRACT FROM PUBLISHER]

In this article, we introduce a class of self-exciting threshold integer-valued autoregressive models driven by independent Poisson-distributed random variables. Basic probabilistic and statistical properties of this class of models are discussed. Moreover, parameter estimation is also addressed. Specifically, the methods of estimation under analysis are the least squares-type and likelihood-based ones. Their performance is compared through a simulation study. [ABSTRACT FROM AUTHOR]

The authors consider a stationary integer-valued autoregressive process of the first order with negative binomial marginals (NBINAR(1)). A set of estimators are considered and their asymptotic distributions are derived. Some numerical results of the estimates are presented. Also, the authors discuss a possible application of the process. [ABSTRACT FROM AUTHOR]

In this article we consider geometric ergodicity and likelihood-based inference for linear and nonlinear Poisson autoregression. In the linear case, the conditional mean is linked linearly to its past values, as well as to the observed values of the Poisson process. This also applies to the conditional variance, making possible interpretation as an integer-valued generalized autoregressive conditional heteroscedasticity process. In a nonlinear conditional Poisson model, the conditional mean is a nonlinear function of its past values and past observations. As a particular example, we consider an exponential autoregressive Poisson model for time series. Under geometric ergodicity, the maximum likelihood estimators are shown to be asymptotically Gaussian in the linear model. In addition, we provide a consistent estimator of their asymptotic covariance matrix. Our approach to verifying geometric ergodicity proceeds via Markov theory and irreducibility. Finding transparent conditions for proving ergodicity turns out to be a delicate problem in the original model formulation. This problem is circumvented by allowing a perturbation of the model. We show that as the perturbations can be chosen to be arbitrarily small, the differences between the perturbed and nonperturbed versions vanish as far as the asymptotic distribution of the parameter estimates is concerned. This article has supplementary material online. [ABSTRACT FROM AUTHOR]