Quasi-likelihood Statistical Inference For The Binomial AR(1) Model

In this paper,we study the quasi-likelihood inference for the binomial AR(1) model which is defined as follows:let δ∈(0,1),ρ∈[max{-δ/1-δ,-1-δ/-δ),1],define β δ(1-ρ),及α=β+ρ,for n∈X,the binomial AR(1) model Xt satisfy the equation, where all the thinning operator are independent.Let θ=(η,u),whereη=ρ(1-ρ)(1-2δ),u=nβ(1-β),then we have uθ(Xt|Xt-1)=Var(Xt|Xt-1)=ηXt-1+u,Furthermore,we denote nβ△=λ,then E[Xt|Xt-1]=ρXt-1+λ,thus,we only need to estimate τ=(ρ,λ)’According to the quasi-likelihood method,we have the following estimation equa-tions:First,we suppose θ is known,and let τ be the solution for the above equation, then we have whereTheorem1For the estimator τ,as T�'∞,we have (?)T(τ-τ)�'LN(0,T-1(θ)),where�'Ldenotes convergence in distribution, At last, we compare the proposed quasi-likelihood estimator with the Yule-Walkerand CLS estimators, we see that the Bias and MSE of the quasi-likelihood are muchsmaller than those for the Yule-Walker and CLS, which denote that our method isacceptable in practice.