Research On Some Parametric And Semiparametric Integer-valued Autoregressive Models

Time series analysis is widely used in actuarial science,environmental pollution control,business management,epidemiology and other fields.It studies a group of real data and reflects the statistical indicators of a certain phenomenon.Therefore,time series is the change law of a certain phenomenon,for example,it can be used to monitor the illegal use of medical insurance cards,understand the dynamic development mechanism of financial products,or predict the rainfall of a certain area in the coming months.This is also a subject that is good at explaining natural and social phenomena.Which puzzles geophysicists,with the rise of land and ocean temperatures,is there an increasing number of annual hurricanes in the North Atlantic and East Pacific basins,or is the intensity of a single storm increasing? A well-designed time series model can give a convincing explanation.Integer-valued time series data is widely exists in production and daily life.For example,the number of blocks newly infected with a certain virus every week in a area for a period of time,the number of criminals in a certain city every month for several years,or the transaction frequency of a stock in a period of time.Compared with continuous valued time series data,the modeling of integer-valued time series data has different challenges.In order to better process and analyze the data,many novel integer-valued time series models have emerged in recent years.As a classical model in integer-valued time series analysis,the integer-valued autoregressive model has a strong vitality since it appeared.There are two ways to generalize this kind of model: one is to use appropriate operators based on different types of real data,for example,binomial thinning operator,negative binomial thinning operator,rounded operator and other operators with different data generative mechanism;second,different types of innovation are used to better capture the disturbance of the data.However,dealing with some integer-valued time series data with special characteristics,a more suitable model is still valuable.To predict the time series with remarkable fluctuations,capture dispersed features of counting time series more accurately,or deal with the natural numerical data with negative correlations,we extend some integer-valued autoregressive models.This thesis is divided into the following three parts:1.Random environment binomial thinning integer-valued autoregressive process with Poisson or geometric marginal.To predict time series of counts with small values and remarkable fluctuations,an available model is the r states random environment process based on the negative binomial thinning operator and the geometric marginal.However,we argue that the aforementioned model may suffer from the following two drawbacks.First,under the condition of no prior information,the overdispersed property of the geometric distribution may lead to great fluctuation of the predictions.Second,because of the constraints on the model parameters,some estimated parameters are close to zero in real-data examples,which may not objectively reveal the correlation relationship.For the first drawback,an r states random environment process based on the binomial thinning operator and the Poisson marginal is introduced.To overcome the second drawback,we propose a generalized r states random environment integer-valued autoregressive model based on the binomial thinning operator to model fluctuations of the data.Yule-Walker and conditional maximum likelihood estimates are considered and their performances are assessed via simulation studies.Finally,in real data applications,we analyzed two groups of urban crime data,compared with the existing random environment model,the performance of the two kinds of random environment model we proposed is better.2.A new extension of thinning integer-valued autoregressive models for count data.The thinning operators play an important role in the analysis of integer-valued autoregressive models,and the most widely used one is the binomial thinning.Inspired by the theory about extended Pascal triangles,a new thinning operator named extended binomial is introduced,which is a general case of the binomial thinning.Compared to the binomial thinning operator,the extended binomial thinning operator has two parameters and is more flexible in modeling.Based on the proposed operator,a new integer-valued autoregressive model is introduced,which can accurately and flexibly capture the dispersed features of counting time series.Two-step conditional least squares estimation is investigated for the innovation-free case and the conditional maximum likelihood estimation is also discussed.We have also obtained the asymptotic properties of the two-step conditional least squares estimators.In real data applications,we analyzed three groups of overdispersed or underdispersed data,and the proposed model is still competitive with many existing parametric models.3.Semiparametric integer-valued autoregressive models on Z.In the analysis of real integer-valued time series data,we often encounter negative values and negative correlations.For integer-valued autoregressive models,there are many parametric models to choose,but some of them are relatively complex.With little information about background of real data,we hope that a simple and effective semiparametric model can be used to obtain more information that usually can not be provided by parametric models,such as the confidence interval of the innovation distribution.But the existing semiparametric model based on thinning operators can only deal with the non-negative data with positive correlation coe cients.In addition,it also has two drawbacks: First,an initial distribution of the innovation is required,but different initial values may lead to different results;Second,the confidence interval of the innovation distribution is not available,which is essential in low-valued data.To overcome these drawbacks,we propose a rounded semiparametric autoregressive model with a log-concave innovation,which can deal with Z-valued time series with positive or negative autoregressive coe cients.The consistences of the estimates for parametric and non-parametric parts of the model are also discussed.In real data analysis,we discuss three groups of data: the first is non-negative integer-valued stock trading data,we compare the proposed model with the existing semiparametric model;the second is stock trading data on Z,we compare some existing parametric model;the third group is batch chemical process data with negative correlations,most of the existing integer-valued autoregressive models can not be used,our model still performs well.