Jin Hong YOU, GeMai CHEN, Min CHEN, Xue Lei JIANG(4)
应用数学学报(英文版). 2003, 19(3): 363-370.
Consider the partly linear regression model y_i = x'_iβ + g(t_i) + ε_i, 1 ≤ i ≤ n, where y_i's are responses, x_i = (x_i1,x_i2,…,x_ip)＇ and t_i ∈ Τ are known and nonrandom design Τ is a compact set in the real line R, β = (β_1, …, β_p)' is an unknown parameter vector, g(·) is an unknown function and {ε_i} is a linear process, i.e., ε_i = ∑ from j = 0 to ∞ of ψ_je_i-j, ψ_0 = 1, ∑ from j = 0 to ∞ of |ψ_j| < ∞, where e_j are i.i.d. random variables with zero mean and variance σ_e~2. Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {ε_i}. The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {ε_i} are an ARMA process, out result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coefficients of the process. Moreover, our result can be used to construct the asymptotically efficient estimators for parameters in the ARMA error process.
