The Research On The Bivariate Integer-Valued Autoregressive Model And Its Test Problem

The integer-valued time series data is quite common in practice,such as the product quality monitoring data,crime data in a region over a period of time,and the traffic accidents data in a region in the morning or at night.Constructing an appropriate analysis system for this kind of data not only helps to study the real problems,but also helps to improve the theoretical system of time series analysis.In recent years,the research on the field of one-dimensional integer-valued time series has been relatively mature,more researchers focus on the bivariate integer-valued times series modelling,in which the study of the characterization of correlation between series is predominant.In order to make the model better applicable to real data,some researchers generalized the Poisson bivariate integer-valued time series model from different perspectives according to the characteristics of integer-valued time series data in practice.These include the seasonal effect and cyclical fluctuation in time series data,and the over dispersion and finite range that integer valued data have.Although research on modeling binary integer-valued time-series data has attracted extensive attention from many researchers,the non-stationary integer-valued time series modelling in bivariate case is less discussed.In addition,the high proportion of observations taking zero values is another important feature of integer valued data,which has not been studied yet.Therefore,there is still a requirement for research on the generalization of bivariate integer-valued time series models.Meanwhile,the test researches have not been sufficiently explored.As an important part of the statistical inference of models,especially in practical applications,the diagnostic test of models is an important tool to judge whether the model has adequately fitted the data.In addition,the bivariate Poisson distribution is the most common discrete distribution,it is not suitable for all data types,especially when the variance of data is significantly larger than the mean or there are more observations with zero values.Since the goodness-of-fit test is an important tool to check whether the model fits the data adequately,it is particularly important to construct a goodness-of-fit test to identify the deviation of the data from the Poisson distribution in the binary case.Based on the above discussion,we systematically summarize the existing work on bivariate integer-valued time series modelling and model test,the following research has been carried out to address the part of the existing literature that need to be supplemented.Firstly,we studied the modelling of bivariate integer-valued time series with more value of zeros,the zero-inflation bivariate integer-valued autoregressive(ZIP-BINAR(1))model is proposed.The zero-inflation in data is characterized by setting zero-inflation Poisson distribution for innovation term.Compared with the Poisson BINAR(1)model,the complexity of the new model estimation and the number of parameters to be estimated increase.The statistical properties and estimation are the focus of the research.Meanwhile,the simulation part verifies the validity of the estimators.Finally,the proposed ZIP-BINAR(1)model is applied to the bivariate real data set with obvious zero inflated feature,and compared the performance with the Poisson BINAR(1)model,and the result shows that the proposed new model performs better than the Poisson BINAR model.Secondly,we considered the modelling of the non-stationary bivariate integer-valued time series data,for which introduced the external circumstance factors,the bivariate integer-valued autoregressive model driven by circumstance factors(Cu BINAR(1))is proposed.The non-stationarity in the model is reflected in the fact that the parameters in the model change with the different circumstantial states,and the statistical properties of the model indicates that conditional mean,and variance would change over the time.For the parameter estimation,we discussed the performance of Yule-Walker and conditional maximum likelihood estimation.To solve the problem of time consuming of conditional maximum likelihood estimation,some functions are written in C and then optimized,which greatly improves the computational speed.Finally,the performance of the new model is evaluated by two real data sets with significant mean level changes,and we compared the Cu BINAR(1)model with the Poisson BINAR(1)model,the Cu BINAR(1)model outer performs the the Poisson BINAR(1)model.Thirdly,as an important part of the statistical inference of model,we supplementary studied the goodness-of-fit test of BINAR model.The goodness-of-fit test statistic based on the bivariate dispersion index is constructed and its limiting distribution under the original hypothesis is derived.In simulation part,we discussed the performance of test statistic under various scenarios,and the results show that the empirical size of test statistic converge to the given nominal level,and the powers converge to one at a faster rate as the sample size increase.Finally,we performed the goodness-of-fit tests on the real datasets discussed in the ZIP-BINAR(1)and Cu BINAR(1)models,the empirical results validate the validity of the goodness-of-fit test statistic.In summary,The innovations of this paper are mainly reflected in the following aspects:Firstly,no researchers have yet discussed the modelling of bivariate integer-valued time series with zero-inflated characteristic.We considered the bivariate zero-inflated discrete distribution for innovation term to characterize the phenomenon of taking more zero values in series.In this setting,the statistical properties and the estimation problems of model are discussed in detail.Secondly,Based on the environment variable proposed by previous researcher,we studied the construction of the non-stationary bivariate integer-valued time series data,proposed the non-stationary bivariate autoregressive model driven by the circumstance factors.The main idea is to set the realized environmental state series for the model.Depending on the state in which environment is located,the parameters of the model taking different values.In parameter estimation part,the conditional maximum likelihood estimation is optimized for the problem of excessive computational effort.Thirdly,although some researchers have conducted exhaustive studies on the test problem under the univariate integer-valued time series model,there are fewer studies on the bivariate model goodness-of-fit test.Unlike the most existing test studies based on the idea of probability generating function,this paper proposed the goodness-of-fit test based on the bivariate dispersion index,and proved that the test statistic is asymptotically distributed normal distribution under null hypothesis,the specific expressions for the elements in the covariance matrix are also given.