Semi-parametric Regression Models And Estimation Problem - L, ~ Q-mixingale Error Sequence Situation

This paper is concerned with parametric component and nonparametric component estimators in a semiparametric regression model. This model is more flexible and explantory than traditional linear or nonparametric model. So many statistics scholars have studied it extensively and obtained many ideal results.Semiparametric regression model yiⁿ=xiⁿβ+ g(tⁿ_i)+ε_iⁿ, 1≥i≥n , where g is an unknown function on a compact set A in R^p and β is an unknown parameter, (x₁ⁿ,x₁ⁿ)^T, 晻?x_nⁿ,t_nⁿ)^T are fixed design vectors and the radom er-rors ε_iⁿ are assumed to be an L^q-mixingale sequences.Fan[1] investigated nonparametric regression model. , Where ε_iⁿ was assumed to be an L^q-mixingale sequences. Let gn(x) =estimate g(x) , Fan[1] obtained the universal consistency of gn(x) un-der uniform integrability of and other conditions. The author wonders whether these results can be fit for semi-parametric model. So the au-thor mainly does the following works.First, the author proves an infinite series expansion of ε_iⁿ and obtains the mean consistency of gn(t) and βn, The author gets rid of the condition of uni-form integrability of , which is not easily satisfied. These results ' generalize the corresponding results in [l] and [2]. Second, complete convergence of gn(t) and βn are studied, These results have not been investigated by Fan[1] and correctly proved in[2], When random errors are martingale difference sequences.In the end, we propose the weight function, give several examples of L^q-mixingale and point out one error in [2].