| Non-linear integer-valued time series exist widely in many fields of life.This nonlinear structure not only exists in finite range integer-valued time series,such as the number of districts with epidemic cases,the number of police stations with crimes,etc,but also exists in infinite range integer-valued time series,such as insurance claim data,global earthquake data,etc.The unsatisfactory fitting effect often appears if we ignore non-linear characteristics in data when fitting these two types of data.Therefore,the modeling of non-linear integer-valued time series becomes particularly important.In this thesis,we study statistical inference for several classes of non-linear integervalued autoregressive models from different perspectives.On the one hand,we establish three kinds of non-linear binomial autoregressive(BAR)models for finite range integer-valued time series.Firstly,we consider cases where time series are affected by unobservable external factors,and propose two kinds of BAR models with Markovswitching.Secondly,in order to describe time-varying characteristics of 9)in the BAR model,we propose the BAR model with variable upper bound.On the other hand,for infinite range integer-valued time series,considering that data will be missing due to instrument failure or recording errors during the acquisition process,we discuss parameter estimation of the self-exciting threshold integer-valued autoregressive(SETINAR)model under missing completely at random mechanism.Based on the above discussion,we obtain the following research results in four aspects:1.Parameter estimation and prediction for the first-order binomial autoregressive model with Markov-switching coefficientsAs the important model for describing finite range integer-valued time series,the BAR model is widely used in the analysis of linear time series.However,in some practical contexts,time series often have non-linear structure due to the influence of external factors,which are sometimes unobservable.In order to describe the influence of unobservable external variables on observation results,we introduce the Markov-switching model into the BAR model,and propose the first-order binomial autoregressive model with Markov-switching coefficients,namely MS-BAR(2,1)model.We prove strictly stationary and ergodic properties of the model,and discuss the problem of parameter estimation and prediction,and further consider the influence of parameter values on the model structure and estimation effect.Finally,the MS-BAR(2,1)model is applied to the analysis of measles infection data,and results show that the model has excellent fitting effect,and verify the feasibility of the prediction method.2.Parameter estimation and prediction for the first-order Beta-binomial autoregressive model with Markov-switching coefficientsIn each regime of the MS-BAR(2,1)model,setting the thinning coefficients as constants has certain limitations.In practice,they may be affected by other factors and change dynamically over time.In order to make this model more fully applicable to more situations,we propose the first-order Beta-binomial autoregressive model with Markov-switching coefficients based on Beta-binomial thinning,namely the MSBBAR(2,1)model.The theoretical properties such as strict stationarity and ergodicity are studied,and the parameters are estimated by conditional maximum likelihood(CML)method,in addition,we consider the conditional expectation prediction.Furthermore,consistency and asymptotic normality of estimators are demonstrated by numerical simulations,and we use the new model to fit the drunken driving data of Pittsburgh to verify its effectiveness.3.Parameter estimation for the first-order binomial autoregressive model with variable upper boundThe upper bound of the random variable is set to a fixed positive integer 9)in the BAR model.This setup brings lots of convenience for practical applications,but it also encounters some difficulties in explaining some practical problems.For example,in the hotel having 9)independent rooms,let (3represents the number of rooms occupied on the day,and the real meaning of 9)is the total number of rooms available on the day.It could change randomly under the influence of power supply failure,which makes constant 9)cannot describe the variable upper bound of (3well.To solve this problem,we replace 9)with a finite range random variable 9),and propose the first-order binomial autoregressive model with variable upper bound,namely VUBBAR(1)model.The results of strict stationarity,ergodicity and moments are given,and asymptotic properties of estimators are established based on conditional least squares(CLS)and CML estimation,meanwhile,the reliability of estimation methods are verified by numerical simulations.We use the new model to fit the trading data of securities companies to illustrate its good performance.4.Parameter estimation for the first-order self-exciting threshold integer-valued autoregressive model with missing valuesIn the study of infinite range integer-valued time series,the introduction of the SETINAR model makes non-linear characteristics of complete observed data can be reflected.However,incomplete observed data will be collected due to incorrect deletion or equipment failure during data collection.Therefore,for incomplete integer-valued time series,we consider the parameter estimation of the SETINAR(2,1)model under missing completely at random mechanism.We study CLS and CML parameter estimation with no imputation based on incomplete observed data,and further consider three imputation methods,and combine them with CLS and CML method to obtain the parameter estimation of the model.Finally,we employ numerical simulations to verify the feasibility of estimation methods,and the Ne SS algorithm is used to obtain accurate estimates of unknown threshold R. |
