Quasi-Likelihood Estimation Of AR(2) Model Under Ordered Restriction

Order restricted statistical inference is an important area in statistical inference. There has been a considerable amount of development in the field since the 1960's. In this paper, we consider quasi-likelihood estimation of AR(2) model under ordered restriction.Suppose that time series {x_t} satisfy the following Auto Regressive model :where {ε_t} is a white noise sequence with same continuous distribution, and it satisfiesFor this model, we know its stationary area is Let thenSuppose X_-1, X₀, X₁,…, X_n be a random sample from AR(2) model, let F_t-1=σ(X_s,s≤t-1).Because of theorem 2.1, we take the following equation as the criterion function:whereWe consider quasi-likelihood estimation ofα, we useα|^ stands for(α|^₁,α|^₂)'Let G_n^* = 0, whereoptimality Second, we consider quasi-likelihood estimation of a under ordered restriction, that is quasi-likelihood estimation of , we useα|^^* stands for (α|^₁^*,α|^₂^*)', then we can use Analogue of Lagrange multipliers ([10]) and Projection of free estimator, such thatIn short, let C = {α₁≥α₂},α|^⁰ = (α|^₀,α|^₀)', so quasi-likelihood estimation ofαunder ordered restriction isWe useα| stands for the real parameter value ofα.Theorem 1 (The strong consistency of QMLE) Assume that time series {x_t} follows the model (1), Ex_t⁴ < +∞,α∈(?)₀, we haveTheorem 2 (The strong consistency of QMLE under ordered restriction) Assume that time series {x_t} follows the model (1), Ex_t⁴< +∞,α|^∈(?)₀, we have Theorem 3(The asymptotic normality of QMLE) Assume that time series{x_t} follows the model(1), Ex_t⁴ <∞,α|^∈(?)₀, we havewhere IAt last, We consider this test problemwhere H₁ :α₁≥α₂ .Quasi-likelihood ratio test static isthe logarithm quasi-likelihood ratio test static isHere, in this paper the logarithm ratio static is defined asTheorem 4 Suppose that H₀ , for (?)t > 0, we have...