Research On Modelling And Control Charts Of Integer-valued Time Series Based On Random Binomial Thinning Operator

Time series analysis is an important branch of statistics,which has a wide application in daily production and life.The well developed models and theories provide great convenience for people to explore and solve practical problems,making time series analysis becomes one of the most used statistical methods in all walks of life.With the deepening of research,it is found that the traditional time series models are very suitable for continuous data,but have obvious defects when analyzing integer-valued data,because their poor explanation of the discrete jump,as well as the unsatisfactory fitting and prediction results.Therefore,since 1980 s,the modeling and statistical inference of integer-valued time series have been paid more and more attention.In economy,finance,industry,environment and medical treatment,integer-valued time series are very common,and their generation mechanisms are diverse,as well as the dependent structures are different.Considering the time series data that could be affected by some external factors,we propose two new classes of INAR processes based on random Binomial thinning operator,and discuss the statistical inference method of these models.In addition,we study the control charts design for a class of zero-inflated Geometric INAR processes.The research of this thesis can provide more theoretical basis and choices for the practical application of integer-valued time series.The main content of this thesis is divided into the following three parts:1.A class of integer-valued Binomial autoregressive model with dependent random coefficients.For integer-valued time series with upper limit,such as the number of rainy days in a week of an city,the Binomial autoregressive(BAR)process is the main modelling method.Due to the influence of many factors such as environment and climate,the autoregressive coefficient in the model often changes with time varying.Therefore,the random coefficient BAR process has attracted much attention.Furthermore,there may be a certain dependence between the two autoregressive coefficients in random coefficient BAR(1)process because of some common factors.This thesis use two-dimensional Normal distribution to flexibly describe this dependence,and propose a class of BAR processes with dependent random coefficients.We prove the strict stationary ergodicity of the model,and discuss the probabilistic and statistical properties of the model,such as transition probability,expectation and variance.The estimation of model parameters is studied by using conditional maximum likelihood method and conditional least square method,and the large sample property of estimator is established.The effectiveness of the estimation method is verified by numerical simulations,and the rainfall data of Shanghai and Hamburg are fitted by the proposed model.The empirical analysis shows that our model has better fitting results,and could be very competitive in practical application.2.A class of covariates-driven mixed Binomial random coefficient integer-valued autoregressive process.In practical problems,the randomness of autoregressive coefficient of INAR process is often caused by some observable explanatory variables.For example,the number of companies with stock transactions is related to variables such as changes in stock prices,the number of criminals is affected by temperature,and so on.Therefore,people use the method of logistic regression to fit the autoregressive coefficient and study the covariates-driven integer-valued autoregressive process.This thesis introduces covariates into a class of mixed random coefficient integer-valued autoregressive process and considers a new model.We give the conditions for the existence of strictly stationary and ergodic solution,and discuss the Markov property and conditional moments of the model.Under the assumption that the innovations follow Poisson distribution and Geometric distribution,the estimations of model parameters are given by conditional maximum likelihood method,and the consistency and asymptotic normality of the estimators are proved.Furthermore,the variable selection for the model is also discussed.We designed a series of numerical simulations to show that the estimate can achieve good results for different models and parameters.The real data analysis also verifies the advantages of our proposed model.3.Control chart for a class of zero-inflated Geometric integer-valued autoregressive process with random coefficient.Statistical process control(SPC)is an important management technology in product quality control and design.Control chart is one of the main tools of SPC.Due to the complex internal dependence structure of integervalued time series,the traditional assumption of observation independence in control chart is no longer applicable.Therefore,designing more scientific and reasonable control charts for various INAR processes has become a hot research topic and has also made rich theoretical and application achievements at present.However,the existing literatures mainly discuss the parameter drift detection of Poisson INAR process,and there are few studies on other models.This thesis analyzes the SPC for the zero-inflated Geometric INAR process based on a class of random Binomial thinning operator.We discuss the CUSUM control chart to detect the upward drifts of the process mean and autocorrelation,construct the monitoring statistics,and give the method to determine the design parameters.The effectiveness of CUSUM control chart is verified by numerical simulation.and the real data analysis results of a group of drug crime data show that our method can monitor the process more effectively,compared with the traditional Shewhart control chart and combined jumps control chart.