Statistical Analysis For Integer-valued Time Series And Asynchronous Data

This thesis focuses on the statistical analysis of two types of integer-valued time series as well as the asynchronous data in continuous-valued time series and longitudi-nal data.First,a first-order integer-valued autoregressive（INAR（1））model driven by moderate deviations from a unit root coefficient is proposed and the conditional least squares（CLS）estimation of the autoregressive coefficient and its asymptotic distribu-tion are obtained.Secondly,a first-order periodic self-exciting threshold integer-valued autoregressive（PSETINAR（2;1,1）_T）process is used to model the integer-valued time series with threshold and period,and a modified quasi likelihood（MQL）estimation approach is used to estimate the parameters and threshold vectors when the distribu-tion of error terms is unknown.Again,a nonuniform first-order autoregressive（AR（1））model with explanatory variables is proposed,considering the effects of synchronous and asynchronous explanatory variables on time series,and the model parameters are estimated.Finally,in the varying coefficient model,the effects of synchronous and asynchronous longitudinal covariates on the longitudinal response variables are ex-plored,the time-varying coefficients are estimated and simultaneous confidence bands（SCB）are constructed for them.Therefore,the main content of this thesis is divided into four parts as follows:1.Statistical inference of INAR（1）model with moderate deviations from a unit rootIn this thesis,we consider the INAR（1）process with coefficientα_n=1-c/k_n,where c>0 is a fixed constant,{k_n}_n∈Nis a sequence that grows to infinity and satisfies k_n=o（n）,which is referred to the INAR（1）model with moderate deviations from a unit root,its autoregressive coefficient gradually converges to a unit root as the sample length increases,in accordance with the sustained growth trend of some economic data.Some fundamental properties of the model are considered,and CLS estimation ofα_nis obtained,which converges to a normal distribution at rate n^1/2k_n.Simulations provide numerical support for the asymptotic property of the estimator and consider the problem of unit root test.Finally,three datasets that liquor offences（including,but not restricted to,drunken driving,intoxication-related incidents,unlawful alcohol sales,etc.）,crimes against the person and COVID-19,are used to illustrate the wide applicability.2.Quasi likelihood estimation of PSETINAR（2;1,1）_TmodelIn Pereira et al.（2015）and Manaa and Bentarzi（2023）,it is assumed that the error term of the PSETINAR（2;1,1）_Tmodel is a Poisson distribution,to adapt a wider range of data characteristics,we remove the specific distribution and make the model more flexible,such as data that can be used to depict overdispersed and underdispersed counts.In this thesis,the quasi likelihood estimation approach is used for analysis,when thresholds are known,MQL approach is used to estimate the parameters and get their asymptotic properties;when thresholds are unknown,a three-step estimation approach is proposed to estimate thresholds by combining the MQL estimation ap-proach and the nested sub-sample search algorithm.Furthermore,the period number is obtained using the periodogram approach when the period is difficult to discrim-inate.Simulations indicate the superiority of MQL estimation approach.Finally,a dataset about claim counts is presented that illustrates the application of these proce-dures,and we discuss the prediction based on conditional expectation and conditional distribution.3.Statistical inference of nonuniform AR（1）model with synchronous and asyn-chronous explanatory variablesWe propose a nonuniform AR（1）model with explanatory variables and consider the parameters estimation when the time series is affected by synchronous and asyn-chronous explanatory variables.Based on the idea of kernel weighting,three one-step approaches are used to obtain the parameters estimation.The first is the weighted last value carried forward approach:using the closest observations of the explanatory vari-ables up to the current moment and weighting them by time proximity;the second is the half kernel weighting estimation approach,using all observations of the explanatory variables up to the current moment and weighting them by time proximity;the third is the full kernel weighting estimation approach,using all observations of the explanatory variables and weighting them by time proximity.Further,due to the asynchrony of explanatory variables,they may be omitted in modeling,so a two-step approach is used to estimate the parameters:in the first step,we use a nonparametric function instead of asynchronous part,combine Taylor expansion and CLS to estimate the co-efficients of synchronous explanatory variables and autoregressive;in the second step,the residuals are used to estimate the coefficients of asynchronous variables when they are considered.We consider the World Bank and Sepsis datasets to further illustrate the practicality and flexibility of proposed approaches,and discuss their prediction problems.4.Statistical inference of varying coefficient model with synchronous and asyn-chronous longitudinal covariatesWe further consider the estimation of time-dependent coefficients in varying coef-ficient model when the longitudinal response variables are affected by synchronous and asynchronous longitudinal covariates.There are two sets of longitudinal covariates,one observed synchronously with the longitudinal response variables,and the other observed asynchronously with them.In the case of synchronous and asynchronous lon-gitudinal covariates without restrictions,we obtain the estimation of time-dependent coefficients with a convergence rate O(n^1/3)using the one-step kernel weighting ap-proach.If synchronous and asynchronous covariates are assumed to be uncorrelated,a two-step approach is proposed to improve the estimation efficiency of time-dependent coefficients for synchronous longitudinal covariates.In the first step,we regress the response variables on the synchronous covariates.Here,two strategies can be used,one is to remove asynchronous parts by centering,another is to replace them by using a nonparametric function.By reducing the smoothing of time-dependent coefficients,we obtain a nonparametric convergence rate O(n^2/5)of time-dependent coefficients for synchronous covariates.In the second step,we regress the residuals from first step on the asynchronous covariates to get the estimation of time-dependent coefficients for asynchronous covariates.To describe the overall trend of time-dependent coefficients,it is common to construct SCBs for them,however,due to the sparse asynchronous longitudinal structure,it is complicated to derive the extreme value distribution of the estimation functions theoretically.Therefore,the wild bootstrap approach is in-troduced to construct the SCBs for time-dependent coefficients and get well coverage in simulations.Finally,a dataset of Alzheimer’s disease is analyzed to illustrate the utility of proposed approaches.