A NGBINAR(1) Model Based On The FGM Copula

Count data is widely used in economy,finance,insurance,transportation,medical and other fields.There are two current research methods: one is state space modeling based on latent process,the other is thinning operator modeling.The second method is the main research methods,such as INAR(1)model based on binomial thinning operator,NGINAR(1)model based on negative binomial thinning operator,etc.Over-dispersed count data has been paid more and more attention by scholars.Many integer-valued autoregressive models based on different distribution of innovation series have been proposed.At the same time,many scholars have carried out a series of extended research based on NGINAR(1)model,and the research on one-dimensional count data has gradually matured.Then,some integer-valued autoregressive models based on two-dimensional count data have been proposed,for example,BINAR(1)model has been widely used.Similarly,on this basis,the research on count data has made some progress.Some scholars began to apply binary Copula,such as Frank Copula,Normal Copula,FGM Copula and so on.Among them,FGM Copula has better effect and is most widely used.The new model proposed in this paper is suitable for dealing with overdispersed two-dimensional count data,the binomial thinning operator in BINAR(1)model is extended to negative binomial thinning operator,and FGM Copula is introduced to construct bivariate geometric distribution.Different from other models,this model applies Copula to the first-order lag term of the model instead of connecting two innovation variables.At the same time,this model is a bivariate generalization of NGINAR(1)model,so it is called NGBINAR(1)model.This model can not only avoid the inapplicability of binomial thinning operator in some cases,but also be suitable for processing dependent two-dimensional count data.In this paper,we first introduce FGM Copula,bivariate geometric distribution,the negative binomial thinning operator and other preparatory knowledge to prepare for the new integer-valued autoregressive model.Then,some statistical properties of the new model are given,including strict stationarity,moment and conditional moment,over-dispersion,etc.,and conditional least squares estimator and numerical simulations are given for the model.The simulation results show that the larger the sample size is,the better the estimation effect become,which illustrates the large sample nature of conditional least squares estimation.Finally,the model is used to fit the crime data of two adjacent communities in Rochester,and the descriptive statistics,sample path diagram,ACF diagram and PACF diagram of the data are given to show its characteristics.It can be concluded that the crime data of adjacent communities have certain correlation and over-dispersion,so the NGBINAR(1)model can deal with the data better in theory.Compared with other models,the fitting results also verify this point,which also show the practical significance of the model.