Research On Some Integer-Valued Time Series Models With Finite Ranges

Integer-valued time series of counts with finite range are widely common in prac-tice,for example,when studying the number of the districts with new cases of a certain infections per week over a period of time reports in finite districts,or studying the number of the occupied workstation per minute reports in finite workstations for a certain university over a period of time.These counts often have the characteris-tics of dependence,discreteness,susceptible to external events,heteroscedasticity and extra-binomial variation and so on.Thus,data with these characteristics is substan-tially different from the traditional real-valued time series data.In addition,data with these characteristics has a natural upper bound n ? N that can never be exceeded,which makes such data different from that taking values in N0.Hence,the first-order integer-value autoregressive(INAR)model and all its extensions applied to infinite-range count processes are not suitable for analyzing these bounded time series counts.Therefore,it becomes more difficult to analyze and model such integer-valued time series with bounded counts.The common models for integer-valued time series with finite range are binomial integer-valued autoregressive(BAR)model,binomial integer-valued ARCH model and binomial integer-valued GARCH model,whose research are originated from the BAR(1)model based on the binomial thinning operator proposed in the 1980s.Later its sta-tistical properties,parameter estimation and statistical inference are established,then the integer-valued BAR(1)models with additive outliers are proposed to analyze the isolated and randomly occurring outliers,binomial integer-valued ARCH model and binomial integer-valued GARCH model are proposed to analyze the volatility(espe-cially the heteroscedasticity).And these models attract much study and generalization once they appear.Based on the characteristics of sensitivity,heteroscedasticity and extra-binomial variation,we provide some integer-valued time series models with finite ranges.The main contents of this thesis contain following three parts.1.Binomial AR(1)processes with innovational outliers.In epidemiology,a sudden change in temperature or environment may lead the new areas with a certain infection disease to surge or sharply decrease,whose occurrence affects the subsequent obser-vations.And the intervention of external measures may lead the new areas with a certain infection to decrease,but it does not immediately reduce to zero.To analyze data with above characteristics,we propose the binomial AR(1)processes with inno-vational outliers.And their perturbation part and underlying process are constructed on binomial thinning operator such that observed data and the generated outliers are integer counts.A main advantage of these models is that the outlier occurs at a certain time and its occurrence affects its subsequent observations.According to the chang-ing size of the outliers,we consider the following two models.The first one is the BAR(1)model with a positive innovational outlier based on the observation increasing sharply at a certain time.And the second one is based on the observation decreasing significantly at a certain time and it is called as the BAR(1)model with a negative innovational outlier.We study the conditional least squares(CLS)estimation and the conditional maximum likelihood(CML)estimation to estimate the parameters both in perturbation part and the underlying process,give the asymptotic properties of the estimators and conduct a simulation study to illustrate the finite sample properties of the CLS and CML estimators.Last,we use the proposed model to fit an artificial data and a real data,respectively.2.Two classes of dynamic binomial integer-valued ARCH models.In a metapop-ulation with fixed size n,n ? N0,the changing of individual or environment may lead the volatility,especially the heteroscedasticity of the observation sequence.And a main representation of this characteristic is that the survival probability of the in-dividual varies with time.To capture the characteristic of time varying,we propose two classes of dynamic binomial integer-valued ARCH models for the data with finite range.The first class is called as the binomial integer-valued ARCH model with a logit transformation(denoted as logit-BARCH model),which captures time-varying survival probability by the lag of the observed process.Note that the observed process taking values in {0,1.2,...,n} makes the smooth change of the survival probability can not be effectively captured.Based on this reason,we proposed the second class of dynamic binomial integer-valued ARCH model,which is called as the score-BARCH model.This model not only encompasses the advantage of logit-BARCH model,but also exploits the conditional probability to introduce score mechanism of the log-likelihood func-tion achieving the purpose of capturing the smooth change of the survival probability.We prove the strict stationarity and ergodicity of the processes,give some stochastic properties,discuss the conditional least squares estimation and the conditional max-imum likelihood estimation,prove their asymptotic properties of the two estimation methods and use a simulation study to compare the finite sample properties of the two estimation methods.Last,we apply the proposed models to two real datasets.3.A new class of beta-binomial integer-valued GARCH models with extra-binomial variation.Note that a limitation of binomial distribution is that its binomial index of dispersion is equal to 1 such that it can not capture the real dispersion of the data with extra-binomial variation.Most modeling methods of such characteristic are based on the conditional mean process instead of the observation process itself such that the heteroscedasticity of the data cannot be described effectively.To model data with extra-binomial variation and heteroscedasticity,we proposed a new class of beta-binomial integer-valued GARCH models.We prove the stationarity and ergodicity of the observed process and its conditional mean process,give some important statistical properties,discuss the conditional maximum likelihood estimation and its asymptotic property.A simulation study are conducted to illustrate the finite sample properties.Last,we apply the proposed models to three real datasets.