| With a wide use of time series in epidemiology,marketing,insurance and environmental science,integer-valued time series has gained rapid development.In recent years,there has been increasing concern and interest in developing bivariate time series models for researchers.In real life,we often encounter such integer-valued time series such as the number of transactions per minute for two stocks,the monthly number of crimes for two certain areas and the daily number of daytime and nighttime traffic accidents,etc.The difference from the one-dimensional time series is that the dependence between two time series consists of not only the dependency of each marginal sequence,and also the cross dependence between different components of two series.Liu(2012)proposed an Integer-valued Generalized Autoregressive Conditional Heteroskedasticity(INGARCH)model based on the traditional bivariate Poisson distribution.However,this model can only fit the bivariate data with a non-negative correlation.Thus in this paper,we mainly study two classes Possion INGARCH models which both allow the positive and negative cross correlations in the second and third chapters,the structure for the latter case is more flexible and easy to interpret.On the other hand,change point detection has drawn much attention for statistical researchers,however,most of the existing results in the literature have been focused on detecting change points in the time domain.In fact,change points in the state domain exist in diverse areas such as physics,environmental science,economics and finance,etc.Unlike time-domain change point methods,statistical inference over the state domain heavily depends on the conditional density and conditional variance,therefore establishing asymptotic properties for the test statistic is more sophisticated.In the Chapter 4,we will utilize the nonparametric hypothesis test to examine the existence of change points in the regression function by deriving the asymptotic distribution of the test statistic.When the existence of change points is affirmative,we further introduce an algorithm based on the bootstrap method to estimate their number together with their locations,and establish a consistent result on the change point estimation procedure.1.We introduce an alternative bivariate Poisson distribution,the value of its dependency parameter is in the field of real number.We therefore establish the bivariate Poisson INGARCH model based on this bivariate Poisson distribution,which can capture zero,positive or negative cross correlation of two series.We provide the stationarity and ergodicity for this model,obtain the estimates based on the maximum likelihood estimation and establish their consistency and asymptotic normality.2.Based on the bivariate Poisson distribution constructed by a multiplication factor,we put forward a class of bivariate Poisson INGARCH models,these models can still allow positive and negative correlations between two series.We also discuss the stationarity and ergodicity of this kind of models.According to the additivity feature in the log-likelihood function of these models,we will adopt maximization by parts algorithm,modified maximization by parts algorithm to estimate unknown parameters.The consistency and asymptotic normality of the above estimators are established as well.In addition,we utilize a more flexible method based on the R package TMB to estimate the parameters,and this method improves the calculation speed and estimation accuracy.3.Under nonparametric autoregressive models,we will apply an anti-symmetric kernel function to the state domain to examine if the regression is smooth,thus test the existence of change points.We establish the asymptotic properties of the test statistic and use the bootstrap method to improve the convergence rate.When change points in the state domain have been detected,we propose an algorithm to estimate their number and locations,and also prove the corresponding convergence result of estimators. |
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