Two Kinds Of Over-dispersed Integer-valued Autoregressive Models And Zero Modification Test

The widespread existence of integer-valued time series has attracted much research,and over-dispersion(variance greater than the mean)is one of the common phenomena in such data.The main causes of over-dispersion are zero-inflation,heavytailedness,and large fluctuations.Since the classical first-order integer-valued autoregressive(INAR(1))——Poisson INAR(1)models,are not suitable for fitting overdispersed data,the improvement of the classical models has become the direction of many researchers.One is to enhance the adaptability of the model by changing the distribution of the innovations,and the other is to modify the model by changing the thinning operator in the model.In this thesis,we propose two types of models for fitting over-dispersed data,one of which is an INAR(1)model with an one-parameter distribution of innovations by changing the distribution of innovations,and the other is a mixed generalized Poisson INAR model by using two thinning operators.In addition,an intuitive method for testing the zero modified in the INAR(1)model is considered.The main content is divided into three parts as follows:1,An INAR(1)Model with An One-Parameter Distribution of Innovations.In order to solve the problem that the Poisson INAR(1)model is not suitable for fitting over-dispersed data,many scholars have proposed various INAR models that can be used for over-dispersed data.However,most of the distributions of the innovations for these models are derivatives of Poisson distribution,such as zero-inflated Poisson,compound Poisson,double Poisson and generalized Poisson distributions,and these distributions usually have more than one parameter.Recently,a Bell distribution with one-parameter has been proposed.This distribution is not only one-parameter and simple in form,but also belongs to the family of exponential distributions.In addition,the variance of the Bell distribution is larger than the mean,indicating that it is suitable for fitting over-dispersed data.Therefore,in this thesis,the Bell distribution is used in integer-valued time series and an INAR(1)model with the binomial thinning operator is considered.The basic properties of the model are considered,and the model parameters are given by three methods: conditional least squares,Yule-Walker and conditional maximum likelihood,and the consistency and asymptotic normality of the three estimates are presented.Subsequently,the performance of the three estimation methods is analyzed by numerical simulations.Finally,in order to see whether the proposed model is advantageous,two real data examples are analyzed and compared to show that the proposed model is competitive compared to other models.2,Mixed Generalized Poisson INAR(p)Model.In the INAR model,the thinning operator plays a very important role,which determines the process of variable generation in the model.The classical Poisson INAR(1)model uses a binomial thinning operator,i.e.,a Bernoulli process.However,in practice,there are cases where the data generation process is not invariant.Therefore,a singleprocess model may not be appropriate in some cases,so a mixed INAR model becomes a better choice.One of the attractive features of the mixed INAR models is that they can capture the structural changes in the time series process.However,most of the existing mixed INAR models are a mixture of binomial thinning operators and negative binomial thinning operators.In this thesis,we propose a mixed INAR model with a generalized Poisson distribution of innovations based on the quasi-binomial thinning operator and generalized Poisson thinning operator in order to fit the over-dispersed data with structural changes,and establish the first-order and higher-order forms of the model respectively.In addition,this thesis gives a proof of the strictly stationary and ergodicity of the two models,and the parameter estimation is performed by the conditional maximum likelihood method,and the numerical simulation results show that the estimation method is effective.Meanwhile,the prediction of the models is discussed and the effectiveness of the prediction is evaluated by numerical simulations.In addition,a Bayesian model averaging prediction method is introduced to avoid choosing the order of higher order models.Finally,the advantages of the mixed generalized Poisson INAR(p)model over other models are demonstrated by two example analyses.3,An Intuitive Method for Testing The Zero-Modified of The INAR(1)Model.For integer-valued time series,one of the major causes of over-dispersion is zero inflation,i.e.,the number of zeros in the data is more frequent than the probability of zero in a Poisson distribution,and vice versa,called zero deflation.Therefore,it is necessary to test whether the number of zeros in a data set is zero-inflated or zerodeflated,which plays a important role in the subsequent model selection.In fact,there are many methods available to test for zero-modification,such as likelihood ratio test,score test and the test of zero index.However,all of the existing tests rely on asymptotic results,thus may not work for small sample sizes.In this thesis,we provide an intuitive test for the zero-modification test of the INAR(1)model by taking the number of zeros in the data as the test statistic,the approximate distribution of the statistic to a Beta binomial distribution,and giving a modification to the estimator of mean.Through numerical simulations and compared with the test of zero index,it is shown that the proposed method can control the size relatively well,and that the power of the test is similar to that of the test of zero index under given alternative hypotheses.Finally,the test is validated with three examples.