Statistical Inference For Some Classes Of Mixture And Dynamic Coefficient Integer-valued Time Series

Integer-valued time series data are fairly common in various fields such as social science,industry,finance,economy,medicine and ecology,etc.The main feature of these data is their integer-valued structure.If we use the traditional real-valued autoregressive model to fit this kind of data,it will bring large deviations,and the results of prediction are not integer-value.Therefore,many traditional real-valued autoregressive models can not be applied to this kind of data.There are two common models to tackle with these integer-valued time series data.They are integer-valued autoregression(INAR)model and integer-valued GARCH(INGARCH)model.These two models are the counterparts of the real-valued autoregressive model and GARCH model in the aspect of integer-valued time series respectively.The studies of INAR models began with the binomial thinning operators.Common INAR models include Poisson INAR model,negative binomial INAR model,geometric INAR model and binomial INAR(BAR)model,etc.Among them,the BAR model is suitable for modeling finite range integer-valued time series.Later,a Poisson integer-valued GARCH model was proposed for dealing with the heteroscedasticity of integer-valued data.We study the statistical inference for some classes of mixture and dynamic coefficient integer-valued time series in this thesis,mainly involving two aspects.On the one hand,we propose two kinds of mixture INGARCH models based on the common multimodality and overdispersion features of data in practice.On the other hand,for the integer-valued time series with finite range,we propose a BAR(1)model with dynamic coefficient,that is the thinning probabilities are dependent on observed values.We study the statistical inference for the mixture INGARCH models and the dynamic coefficient BAR(1)model.The following three parts constitute the main content of this thesis.1.A generalized mixture integer-valued GARCH model for infinite range time series.Referring to the existing literature,we find that most integer-valued models are obtained by assuming that the time series are driven by a unimodal innovation series.However,many time series may exhibit the multimodality feature either in the marginal or the conditional distribution.To deal with this problem,many researchers have considered the mixture models in the integer-valued time series field.We propose a generalized mixture integer-valued generalized autoregressive conditional heteroscedastic model to provide a more flexible modeling framework.This model includes many mixture integer-valued models with different distributions already studied in the literature.The conditional and unconditional moments are discussed and the necessary and sufficient first-and second-order stationary conditions are derived.We also investigate the theoretical properties such as strict stationarity and ergodicity for the mixture process.The conditional maximum likelihood estimators via the EM algorithm are derived and the performances of the estimators are studied via simulation.The model can be selected in terms of both the number of mixture components and the number of orders in each component by several different criteria.A real data example is also given to assess the performance of the model.2.A mixture INGARCH model for finite range time series.In practice,there are many integer-valued time series with finite range.For example,the number of companies trading at a certain time in finite securities companies,and the number of districts infected at a certain time in finite districts.We apply the flexible mixture modeling method to the finite range integer-valued time series,and establish a binomial mixture autoregressive conditional heteroscedasticity(B-MINGARCH)model,which includes a kind of binomial mixture autoregressive(B-MINARCH)model with strict stationary and ergodicity.The necessary and sufficient first-and second-order stationary conditions are derived.The conditional maximum likelihood estimators via the EM algorithm are derived and the performances of the estimators are studied via simulation.A real data example is also given to assess the performance of the model.3.A BAR(1)model with dynamic coefficient.BAR models are suitable for time series with finite range.The BAR(1)model has been concerned by many scholars.A significant limitation of the BAR(1)model is that the thinning probabilities and at time do not depend on the process up to that time.The thinning probabilities are constant parameters which do not change with time.In fact,they may be affected by environmental factors,or to some extent,change with the change of the observations.Based on the BAR(1)model,we remove the nonrandom restriction of the thinning probabilities,and consider a kind of first-order binomial integer-valued autoregressive model with observation dependent dynamic coefficients and .We show the strict stationary and ergodicity of the process.Three estimation methods of model parameters are discussed: conditional maximum likelihood estimation,conditional least square estimation and maximum empirical likelihood estimation.We prove the consistency and asymptotic normality of the estimators,consider the coverage probability of confidence regions and the empirical likelihood test of parameters.We apply the proposed model to a real dataset.