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In this paper, we consider the problem of testing for a parameter change in Poisson autoregressive models. We suggest two types of cumulative sum (CUSUM) tests, namely, those based on estimates and residuals. We first demonstrate that the conditional maximum likelihood estimator (CMLE) is strongly consistent and asymptotically normal and then construct the CMLE-based CUSUM test. It is shown that under regularity conditions, its limiting null distribution is a function of independent Brownian bridges. Next, we construct the residual-based CUSUM test and derive its limiting null distribution. Simulation results are provided for illustration. A real-data analysis is performed on data for polio incidence and campylobacteriosis infections.

Regularity conditions are given for the consistency of the Poisson quasi-maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer-valued autoregressive (INAR) and integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.

This study analyses 18 years of weekly reported dengue cases (January 2002–December 2020; 988 weeks) from Costa Rica’s Central Valley to examine seasonal and multi-year patterns. To model the spatio-temporal dynamics of dengue, we employ three statistical approaches for case counts: the spatial hurdle integer-valued generalized autoregressive conditional heteroskedasticity (INGARCH) model, the spatial zero-inflated generalized Poisson (ZIGP)-INGARCH model, and the endemic–epidemic (EE) model. Covariates include rainfall and maximum temperature or alternatively seasonal Fourier terms to represent annual seasonality. Using a Bayesian framework, we fit the spatial INGARCH-family models to weekly dengue cases. The EE model and the ZIGP-INGARCH model, both with Fourier seasonal terms, show the best predictive accuracy and provide estimates of seasonal intensity and peak timing relevant for dengue surveillance. Incorporating annual seasonality improves modelling of multivariate weekly dengue cases in Costa Rica’s Central Valley, underscoring the importance of cyclical patterns for strengthening early warning systems and guiding targeted vector control.

This article studies theory and inference of an observation-driven model for time series of counts. It is assumed that the observations follow a Poisson distribution conditioned on an accompanying intensity process, which is equipped with a two-regime structure according to the magnitude of the lagged observations. Generalized from the Poisson autoregression, it allows more flexible, and even negative correlation, in the observations, which cannot be produced by the single-regime model. Classical Markov chain theory and Lyapunov's method are used to derive the conditions under which the process has a unique invariant probability measure and to show a strong law of large numbers of the intensity process. Moreover, the asymptotic theory of the maximum likelihood estimates of the parameters is established. A simulation study and a real-data application are considered, where the model is applied to the number of major earthquakes in the world. Supplementary materials for this article are available online.

This research proposes a comprehensive ALG model (Adaptive Loglinear zero-inflated Generalized Poisson integer-valued GARCH) to describe the dynamics of integer-valued time series of crime incidents with the features of autocorrelation, heteroscedasticity, overdispersion and excessive number of zero observations. The proposed ALG model captures time-varying nonlinear dependence and simultaneously incorporates the impact of multiple exogenous variables in a unified modeling framework. We use an adaptive approach to automatically detect subsamples of local homogeneity at each time point of interest and estimate the time-dependent parameters through an adaptive Bayesian Markov chain Monte Carlo (MCMC) sampling scheme. A simulation study shows stable and accurate finite sample performances of the ALG model under both homogeneous and heterogeneous scenarios. When implemented with data on crime incidents in Byron, Australia, the ALG model delivers a persuasive estimation of the stochastic intensity of criminal incidents and provides insightful interpretations on both the dynamics of intensity and the impacts of temperature and demographic factors for different crime categories.

When dealing with time series with outlying and atypical data, a commonly used approach is to develop models based on heavy-tailed distributions. The literature coping with continuous-valued time series with extreme observations is well explored. However, current literature on modelling integer-valued time series data with heavy-tailedness is less considered. The state of the art research on this topic is presented by Gorgi (J R Stat Soc Ser B (Stat Methodol) 82:1325–1347, 2020) very recently, which introduced a linear Beta-negative binomial integer-valued generalized autoregressive conditional heteroscedastic (BNB-INGARCH) model. However, such proposed process allows for positive correlation only. This paper develops a log-linear version of the BNB-INGARCH model, which accommodates both negative and positive serial correlations. Moreover, we adopt Bayesian inference for better quantifying the uncertainty of unknown parameters. Due to the high computational demand, we resort to adaptive Markov chain Monte Carlo sampling schemes for parameter estimations and inferences. The performance of the proposed method is evaluated via a simulation study and empirical applications.

In this paper, we study a robust estimation method for observation-driven integer-valued time series models whose conditional distribution belongs to the one-parameter exponential family. Maximum likelihood estimator (MLE) is commonly used to estimate parameters, but it is highly affected by outliers. We resort to the Mallows’ quasi-likelihood estimator based on a modified Tukey’s biweight function as a robust estimator and establish its existence, uniqueness, consistency and asymptotic normality under some regularity conditions. Compared with MLE, simulation results illustrate the better performance of the new estimator. An application is performed on data for two real data sets, and a comparison with other existing robust estimators is also given.

This research models and forecasts bounded ordinal time series data that can appear in various contexts, such as air quality index (AQI) levels, economic situations, and credit ratings. This class of time series data is characterized by being bounded and exhibiting a concentration of large probabilities on a few categories, such as states 0 and 1. We propose using Bayesian methods for modeling and forecasting in zero-one-inflated bounded Poisson autoregressive (ZOBPAR) models, which are specifically designed to capture the dynamic changes in such ordinal time series data. We innovatively extend models to incorporate exogenous variables, marking a new direction in Bayesian inferences and forecasting. Simulation studies demonstrate that the proposed methods accurately estimate all unknown parameters, and the posterior means of parameter estimates are robustly close to the actual values as the sample size increases. In the empirical study we investigate three datasets of daily AQI levels from three stations in Taiwan and consider five competing models for the real examples. The results exhibit that the proposed method reasonably predicts the AQI levels in the testing period, especially for the Miaoli station.

Dengue fever is transmitted to humans through the bite of an infected mosquito and is prevalent in all tropical and subtropical climates worldwide. It is thus essential to model weekly dengue fever counts and other infectious diseases that exhibit spatial-temporal dynamics, overdispersion, spatial dependence, and a high number of zeros. To address these characteristics, this study introduces a spatial hurdle integer-valued GARCH (INGARCH) model and an improved version of the spatial zero-inflated generalized Poisson (ZIGP) INGARCH model with and without meteorological variables. Implementing two parameters in the distance function influences the spatial weight between two locations: one controls the decay rate, while the other shapes the decay curve. We employ these newly designed models to analyze time-series counts of infectious diseases - specifically, weekly cases of dengue hemorrhagic fever in four northeastern provinces of Thailand. Applying these models allow us to offer inferences, predictions, and model selections within a Bayesian framework through Markov chain Monte Carlo (MCMC) methods. We then compare models based on the Bayes factors and the mean squared error of fitting errors. The results for the spatial ZIGP INGARCH models are remarkably good, but the spatial INGARCH model incorporating meteorological variables outperforms the other two.

This study proposes a class of nonlinear hysteretic integer-valued GARCH models in order to describe the occurrence of weekly dengue hemorrhagic fever cases via three meteorological covariates: precipitation, average temperature, and relative humidity. The proposed model adopts the hysteretic three-regime switching mechanism with a buffer zone that are able to explain various characteristics. This allows for having consecutive zeros in the lower regime and large counts to appear up in the upper regime. These nonlinear hysteretic integer-valued GARCH models include Poisson, negative binomial, and log-linked forms. We utilize adaptive Markov chain Monte Carlo simulations for making inferences and prediction and employ two Bayesian criteria for model comparisons and the relative root mean squared prediction error for evaluation. Simulation and analytic results emphasize that the hysteretic negative binomial integer-valued GARCH model is superior to other models and successfully offers an alternative nonlinear integer-valued GARCH model to better describe larger values of counts.