| Integer-valued time series exist widely in people’s daily production and life,such as the number of people hospitalized due to a certain epidemic every day,the number of weekly claims paid by an insurance company for a certain policy,the number of major earthquakes occured around the world each year,and so on.Such data generally have strong discreteness and dependence.Because this kind of data is very different from the time series data with real value,the traditional time series model cannot fit it effectively.In recent years,the modeling of integral-valued time series data has been a hot topic in the academic field.Currently,there are mainly two types of modeling methods for integer-valued time series data:(a)Integer-valued autoregressive model based on sparse operator;(b)State space model based on latent process.As these two types of models have been widely used to analyze and process integer-valued time series data in various fields,statisticians are increasingly finding out two problems that are hard to solve.On the one hand,when the order of the sequence is very large,the model will inevitably contain some non-significant orders,which will not only increase the complexity of the model,but also affect the statistical inference results of the model.On the other hand,these two types of models do not consider the influence of external factors on the sequence.In many practical cases,the observed values of integer-valued time series may be affected by some exogenous variables at different moments,and these influences cannot be ignored.Because the structure of the integer-valued time series model is very complex,so it is very difficult to determine the order of the model and introduce exogenous covariables.The common order determination method of time series models is AIC criterion function or BIC criterion function,but many statisticians point out that the results obtained based on AIC criterion function and BIC criterion function are unstable,and such biased results will inevitably affect statistical inference.In addition,some statisticians proposed to study the relationship between exogenous covariates and integervalued time series data by introducing exogenous covariates into the error term of the integer-valued autoregressive model.Although this method is simple and feasible,it is very difficult to accurately select the exogenous variables related to the observed values.In general,a variety of exogenous variables are introduced into the model in order to analyze the relationship between the data in detail.However,this approach inevitably introduces some unnecessary exogenous variables into the model.If the model contains unnecessary exogenous variables,it will not only reduce the accuracy of the model parameter estimation,but also cause the deviation of the predicted value of the model.In order to better solve the modeling problem of integer-valued time series data,the influence of exogenous covariables on serial observations is considered in this paper,and the order determination and variable selection of the proposed model are studied based on the penalized estimation method.The main content of this paper is the following three parts:First,the order determination problem of integer-valued generalized autoregressive conditional heteroscedasticity(INGARCH)model.As is known to all,when the INGARCH model is used to model high-order integer-valued time series data,it is inevitable that some non-significant orders will be included in the model.Traditional estimation methods(such as conditional maximum likelihood)cannot strictly estimate the coefficients of these orders to zero.Including these orders in a model will not only increase the complexity of the model,but also negatively affect the results of statistical inference.In order to solve this problem,we propose a penalized conditional maximum likelihood(PCML)method for the structure of the INGARCH model based on penalized estimation method to delete the non-significant orders in the model and estimate the parameters of significant orders at the same time.With the help of theoretical method and numerical simulation,we prove that the PCML estimator is consistent and Oracle properties.At the same time,we use PCML method to analyze the number of Campylobacter infection cases in northern Quebec,Canada,and the analysis results show that our method is also effective in dealing with practical problems.Second,the variable selection problem of INGARCH(1,1)model with covariables.In order to better study the relationship between external factors and integer-valued time series observations,Agosto et al.(2016)further extended the INGARCH model and proposed an INGARCH model with covariables.When this model is used to analyze a practical problem,it is not possible to know in advance which external factors will significantly affect the observed values of the integer-valued time series,so a variety of covariables are usually introduced into the model.This may lead to a large number of non-significant covariables in the model,which will affect the results of statistical inference.To solve this problem,we propose a new penalized estimation method,which can delete the non-significant covariates in the model and simultaneously estimate the parameters of the model.Through theoretical research,the limit properties of the penalized estimator are given,and the conclusion is proved to be correct by numerical simulation.Third,the statistical inference of INAR(GINARS)model with covariables based on generalized power series sparse operators.In order to improve the applicability of GINARS model,we introduce covariables into GINARS(1)model and propose GINARS(1)model with covariables.This model can be used to study the relationship between non-stationary integer-valued time series data and external factors.In order to solve the problem of parameter estimation and variable selection of the model,we combine the penalty function with the least square method,and put forward the penalized conditional least squares(PCLS)method.In theory,we study the large-sample properties of PCLS estimator.In addition,we use numerical simulation to test the effectiveness of PCLS method in parameter estimation and variable selection.The simulation results show that the PCLS method has excellent performance. |
