Modeling And Empirical Likelihood Inference For Bivariate Integer-valued Autoregressive Process Based On Binomial Thinning Operator

In recent years, there has been a growing interest in integer-valued time series. In real applications researchers often have to cope with multivariate count data. Therefore, it seems important of modeling that kind of the data. For the bivariate situation, existing results focus on bivariate Poisson and bivariate Negative Binomial marginal. Then make statistical inference for the modeling data. We often use maximum likelihood estimation when estimating the data of the model, which should know the distribution family of the data. Empirical likelihood (EL) is a nonparametric method of inference based on a data-driven likelihood ratio function. It does not require us to specify a family of distribution for the data. Meanwhile, it straightforwardly forms the confidence regions for parameters, then give us the maximum empirical likelihood estimators, which has wide applications. Moreover, when handling the bivariate autocorrelated count data, we find that there is a class of data which (0,0) can not hold at the same time((0,0) truncation). Supposing that we still use a model with bivariate Poisson or bivariate Negative Binomial marginal, the estimation results will be poor with some potentially significant statistic findings probably missing, which lead to erroneous conclusions and bring uncertainty to the research and applications. Therefore, it’s necessary to construct a new model so as to fit for the data.In this paper, we give the model and empirical likelihood inference for the bivariate integer-valued autoregressive process based on the binomial thinning operator. There are mainly three sections. First, we give the empirical likelihood ratio statistics for existing bivariate autoregressive model, and derive the limit distribution, Construct the confidence regions for the parameters. Then make the simulation using bivariate Poisson and bivari-ate Negative Binomial distributions as the innovation. Meanwhile, comparing with the EL method, we construct the confidence regions for the parameters based on the normal ap-proximation method. Second, we give the maximum empirical likelihood estimators for the model in the first section, prove the consistency and asymptotic normality. Then make the simulation derive the maximum empirical likelihood as well as conditional least squares and conditional maximum likelihood estimators in order to compare with each other. Third, we propose a new stationary bivariate first order mixed integer-valued autoregressive process with zero truncated Poisson marginal distribution. Some properties about this process are considered, such as probability generating function, autocorrelations, expectations and co-variance matrix under conditional and unconditional situation. We also establish the strict stationarity and ergodicity of the process. Estimators of unknown parameters are derived by using Yule-Walker, conditional least squares and maximum likelihood methods. The perfor-mance of the proposed estimation procedures are evaluated through Monte Carlo simulations. An application to a real data example is also provided.First of all, we will introduce an important thinning operators and two kinds of bivariate distributions,(ⅰ) the binomial thinning operator "o" The binomial thinning operator is defined as α o where X is an non-negative discrete random variable; a E [0,1);{Yi} is a sequence of i.i.d. random variables with Bernoulli(α) distribution, i.e., P(Yi= 1)= 1-P(Yi= 0)= α.(ⅱ) bivariate Poisson distribution A non-negative integer-valued random vector (X, Y)T is said to follow a bivariate Poisson distribution if the joint probability mass function is given by where s= min(x,y), λ1> 0, λ2> 0,φ ∈ [0,min(λ1, λ2)). Denote this distribution as BP(λ1,λ2,φ).(iii) bivariate Negative Binomial distribution A non-negative integer-valued random vector (X, Y)T is said to follow a bivariate Negative Binomial distribution if the joint probability mass function is given by where λ1> 0, λ2> 0,β> 0. Denote this distribution as BVNB{λ1, λ2,β).Now we will introduce the main results of this paper.1. The confidence regions of the parameters for the bivariate first-order integer-valued autoregressive process (BINAR(1)) based on empirical likelihood method.Proposed by Pedeli and Karlis (2010), the bivariate sequence{Xt} is said to be BINA-R(1), if where(i) A is a 2 x 2 diagonal matrix with independent elements;(ii) " Ao" is a matrical operation which acts as the usual matrix multiplication keeping in the same time the properties of the binomial thinning operation;(ⅲ){εt}is a i.i.d.non-negative integer-valued bivariate random vector.Let E(εjt)=λj,j=1,2；E(ε1tε2t)=φ+λ1λ2,suppose they are all limit,then E(Xht|Xj,t-1)= αjXj,t-1+λj,j=1,2；Cov(X1t,X2t |X1,t-1,X2,t-1)=Cov(ε1t,ε2t)=φ.In order to make empirical likelihood inference,we should make{Xt].satisfy the follow-ing conditions,(a){Xt)has the strict stationary and ergodicity process;(b)‖Xt‖6<∞.Based on the conditional least squares(CLS)estimation,let Q(θ)= where θ=(α1,α2,λ1,λ2,φ)T.Then by using =0,we can derive CLS estimator θCLs.And we can prove that n�'∞.Based on the theory of the empirical likelihood inference suppose by Owen(1988),the empirical likelihood ratio function of BINAR(1)is The maximum may be found via Lagrange multiplier method. Let S: where γ and b are the Lagrange multiplier. pt/1-nbT(θ)Dt(θ)+γ=0,we can derive that 0: n+γ,then we have pt where b(θ)satisfies =0.then the ELR statistics is Now we derive the limit distribution,the confidence region and the hypothesis testing of L(θ) through following theorems.Theorem 1 Under (a)and (b),we have L(θ)�'LX2(5),n�'∞Theorem 2 For the parameter θ,the 100(1-δ)% confidence region is IEL={θ|L(θ)≥r}, where r=exp{-1/2X2,1-δ(5)}.Theorem 3 Suppose Ho:θ=θ0←�'H1:θ≠θ0,where θ0 is the true value of θ. Then under the null hypothesis H0,for the significance level 0<δ<1,the rejection region is W={θ|L(θ)≤exp{-1/2X2,1-δ(5)}}.Finally,we study the coverage probability of the confidence region for the parameters based on the normal approximation and EL methods when make{εt)be BP(λ1,λ2,φ)and BVNB(λ1,λ2,β).From the results of the simulation,we find that both of the coverage. probability of the confidence regions increase with the increase in sample size n and tends to the confidence level, thus the two methods are acceptable. Moreover, the confidence region for EL method is superior to which for NA method.2. The maximum empirical likelihood (MEL) estimation of the parameters for BINAR(1) process based on the binomial thinning operator.From the ELR statistics of BINAR(1), we can derive as follows, the log EL function Le(θ) then the MEL estimator can be defined Let G where θ0 = (α10,α20, λ10, λ20,φ0)T is the true value. Next we prove the limit properties of the estimators, that is consistency and asymptotic normality.Theorem 4 Under (a) and (b), when n�' ∞, then θMEL is consistent, and we have where Ω2(θ0) and V-1 are defined in Theory 3.1.1. in Chapter 3.Then we make the simulation derive the maximum empirical likelihood as well as con-ditional least squares estimators θCLS and conditional maximum likelihood estimators θCML of BINAR(1) process using MATLAB when{εt} is BP(λ1,λ2,φ) or BVNB(λ1,λ2,β), and compute the bias (Bias) and standard error (SE) of the estimators. From the results of the simulation, we can see that as the sample size n increasing, the Bias and SE of θMEL, θCML and θCLS decrease, which means all the three estimations are acceptable, especially for the large sample. Meanwhile, we can see that SE{θMEL) is smaller than SE(θCLS) and SE{θCML) when n= 500, which means 9MEL is better than θCLS and θCML. Finally it makes great applications for the EL estimation through the real data analysis.3. Bivariate first-order mixed integer-valued autoregressive process with zero truncated bivariate Poisson marginal based on the binomial thinning operator (ZTBPINAR(1)).First we should define the zero truncated bivariate Poisson distribution.Definition 1 A non-negative integer-valued random vector (X, Y) is said to follow a bivariate zero truncated Poisson distribution if the probability mass function is given by where (i,j) ∈ N02\(0,0)}, s= min(i,j), λ1> 0, λ2> 0 and φ ∈[0,min(λ1,λ2)). Denote it as ZTBP(λ1,λ2,φ).Definition 2 A bivariate random variable sequence{Xt} is a stationary ZTBPINAR(1) process, if where(ⅰ) Xt= (X1t,X2t)T follows ZTBP(λ1,λ2,φ);(ⅱ) " o" is the binomial thinning operator defined above;(ⅲ) A= diag(α1, α2) with αi ∈ [0,1), i = 1,2;(iv){εt}为i.i.d.non-negative random variable sequence, independent with the sum sequence {Yi}in"o" and {Xt-l}(l≥1).From the definition and the stationarity of the process, we can derive the probability generating function (PGF) of{εt} through the relationship between{εt} and{Xt}, then the joint probability mass function and 2-step ahead moment of{εt} can be given. Next we study the statistical properties of the process ZTBPINAR(1), such as the mean, the variance,the h-step ahead conditional mean, h-step ahead conditional variance of{Xt} and so on. Meanwhile, we prove that this process is unique strict staitonary and ergodic.Theorem 5 There exists a unique strict stationary integer-valued random series{Xt} satisfying ZTBPINAR(1) and Cov(Xs,εt)=0 for s< t. The unconditional first and second moments of that strict stationary series{Xt} exist. Furthermore, the process is an ergodic process.Then we discuss the parameter estimations of ZTBPINAR(1) process. Three estima-tion methods are considered, Yule-Walker (YW) estimation, conditional least squares (CLS) estimation and conditional maximum likelihood (CML) estimation. Next we prove the limit properties of YW and CLS estimators.From the 2-step ahead moment of{Xt}, we can construct the equation of the parameters, where. Therefore, the YW estimator is the solution of that equation above which can be easily derived by using numerical method. Then we discuss the limit properties of θYW.Theorem 6 Assume that E‖Xt‖2< ∞, E|X1t·X2t|< ∞, then θYW is α strongly consistent estimators for θ.Prom the conditional mean and covariance of {Xt}, we can construct the function, by minimizing which can derive CLS estimators 9CLS. That isThen we discuss the limit properties of LS.Theorem 7 Assume that E|Xjt|< ∞, j = 1,2 and E|X1t2 · X2t2|< ∞, then the CLS estimators θCLS is consistent. Moreover, as n �' ∞, we have where W and V are defined in Theory 4.2.2. in Chapter 4.From the conditional distribution of{Xt}, we can construct the conditional likelihood function of θ, where f(Xt|Xt-1,α1,α2, λ1,λ2,φ) is the conditional distribution of{Xt}. Then the CML estimators are obtained by numerical method through maximizing the above log-likelihood function, that isThen we conduct some simulations to verify our proposed methods. From the results, we can see that as the sample size n increasing, the Bias and SSE of all estimators decrease, which shows that all three methods are reliable, and can give good estimators, especially for large sample sizes. For the YW and CLS, the estimators α1 and α2 are always underestimated. Moreover, the SSE of CML is much smaller than those of YW and CLS, which indicate that the CML is more efficient in practice. Although the CML requires most calculation time among the three methods.