Statistical Inference For Three Types Of Integer-valued Autoregressive Models

As an important branch of time series,integer-valued time series models have a wide range of applications,for example,the lynx data of Canadian with nonlinear structure,the weekly rainy days in a fixed area with finite range of values and the difference in scores between two football matches with both positive and negative integer-values.Based on the above three types of integer value time series data,this paper discusses the following three important problems of integer value autoregressive model: This thesis mainly discusses three important problems of the integer value autoregressive model for the above three types of integer value time series data: First,an integer value autoregressive model with two-threshold-variable is proposed to extend the integer value threshold model through the product of indicator function,which provides a new idea for modeling the distributed nonlinear structure integer-valued time series data;Secondly,the exchangeable Conway-Maxwell-Poisson-Binomial(CMPB)thinning operator is established,and the CMPB integer-valued autoregressive model is proposed to analyze the bounded non-negative integer-valued time series data composed of n exchangeable and dependent units;Finally,the trinomial difference thinning operator is constructed,and the trinomial difference autoregressive model is proposed,which provides a way to solve the problem that the counting sequence does not include-1when dealing with Z-value time series data.Therefore,the main content of this thesis is divided into the following three parts.1.Two-threshold-variable integer-valued autoregressive model.When dealing with nonlinear data such as the southern oscillation index in meteorology,the quarterly gross national product in the United States and the daily number of trades of a stock financial and economic fields,characterizing these data often requires the use of multithreshold variable.In view of the problem that the classical integer-valued threshold autoregressive models basically contain only one threshold variable.This thesis establishes an integer-valued autoregressive model with two-threshold-variable based on binomial thinning operator,and the state space is divided by endogenous variables.The whole space is divided into four subspaces,and the classical model is extended by the product of indicator function.The model distinguishes innovations in different regions and the threshold parameters are determined by considering lots of past information,which makes the model more flexible,practical and reliable.In this thesis,we discuss the basic properties of the proposed model,including the mean,variance,strict stationarity and ergodicity,obtain the estimates of the model parameters by the conditional least squares method and discuss the asymptotic normality of the conditional least squares estimates when the threshold variables are known and unknown,respectively.In order to alleviate the computational pressure,this thesis proposes a reasonable method to search for the threshold estimates,and the numerical simulation results are consistent with the theoretical expectations.In addition,this thesis applies the proposed model to two sets of data,namely,the daily number of trades of a stock and the number of trades in 5-min intervals of a stock data and obtains reasonable conclusions by comparing with the previous models.2.Bounded integer-valued autoregressive model based on CMPB thinning operator.Bounded non-negative integer-valued time series data is also one of the data types which scholars are interested in.For bounded non-negative integer-valued time series with n exchangeable and dependent units,the exchangeable CMPB thinning operator is constructed in this thesis based on CMPB distribution.And then the CMPB autoregressive model is proposed,which can realize the analysis of bounded non-negative integer-valued time series data with under-dispersion,equi-dispersion and over-dispersion by adjusting the parameter changes.This kind of data has a strong application value in many fields,such as,the number of alarms in a fixed area and the number of the occupied workstation per minute of a certain university over a period of time,etc.These time series in a fixed range are discreteness,dependence and susceptible to external events,and they often have a natural upper bound that can never be exceeded.In this thesis,the stationarity and ergodicity of CMPB autoregressive model are considered.The consistence and asymptotic normality of conditional maximum likelihood estimators are illustrated by Boxplots and QQ-plots.In addition,the model is applied to the weekly rainy days,and the superiority of the model is further illustrated by comparing the fitting results with other integer-valued time series models.3.Integer-valued autoregressive model based on trinomial difference thinning operator.When time series data exhibit an infinite range of negative,zero,and positive values,such as the zonal annual means temperature and stock daily return data and so on,it is common to deal with them by using rounded function or Z-valued autoregressive models based on the signed binomial thinning operator.However,there are two limitations in these methods,one is that the counting sequence of such methods is composed of 0 and 1,i.e.,the case of-1 is ignored.The second is that the conditional maximum likelihood estimation of such models is difficult to be obtained.Therefore,based on this phenomenon,this paper constructs the trinomial difference thinning operator,and then proposes the trinomial difference Z-valued autoregressive model.There are two innovative points lies in this model.Firstly,the trinomial difference thinning operator’s counting sequence retains i.i.d.and includes-1,i.e.,the counting sequence takes value in {-1,0,1}.Secondly,the incorporated trinomial difference thinning operator makes the conditional maximum likelihood estimate easier.In this thesis,we discusses the two-step conditional least squares estimate and the conditional maximum likelihood estimate,establish their asymptotic properties of the estimators.Finally,we apply the proposed model to the data of the daily difference between the close and open prices in a stock to illustrate the potential application of the model.