Uniformly Asymptotic Normality Of The Regression Weighted Estimator For (?)-mixing Samples

Consider the fixed design regression modelwhere the design points x_n1,…, x_nn∈A, which is a compact set of R^d,g is a bounded real valued function on A,ε_n1,…,ε_nn are regression errors with zero mean and finite variance. The weighted function estimate of g iswhere weight functionω_ni(x), i = 1,2,…, n, depend on the fixed design points x_n1,…, x_nn and on the number of observations n.In the independent case, estimate (2) has been considered by many researchers, such as, Priestly and chao(1972), Clark(1997), Georgiev(1984a,b,1998), Georgiev and Gre-blicki(1986) and the references therein. In various dependence cases, g_n(x) has also been investigated extensively. For example, Fan(1990), Roussas(1989), Roussas et al.(1992), Tran et al.(1986), Yang(1999) and the references therein. under theα-mixing condition, asymptotic normality of estimate (2) has been established by Roussas et.(1992); Yang(2003) has studied the Uniformly asymptotic normality of of estimate (2) under the negative associated condition, and gotten the convergence rate: n^-1/4 However our purpose is to study he Uniformly asymptotic normality, the rate of the uniformly asymptotic normality of estimate (2) and some applications. To get the theoretical result, we assumeAssumption (A1) (1) For each n, the joint distribution of {ε_ni, 1≤i≤n} is the same as that of {ξ₁,…,ξ_n}; (2) {ξ_j,j≥1} is a p-mixing random variable sequence withzero mean and finite second moment; (3)for someDenoteAssumption (A2) (1)Assumption (A3) There exist positive integers p = p(n) and q = q(n) such that for sufficiently large nand as n→∞whereLet is the distributionfunction of N(0,1). Here we will prove the following results.Theorem 1.1 If Assumptions (Al)-(A3) hold, thenNote that implies u(q)→0, we can getCorollary 1.1 If Assumptions (A1)-(A3) hold, thenCorollary 1.2 If Assumptions (A1)-(A3) hold forδ≥2/3,thenFurther, ifω_n = O(n ^-1 ) andλ≥7/6, thenCorollary 1.3 If Assumptions (A1)-(A3) hold for 0 <δ< 2/3, then Further, ifω_n = O(n^-1) andλ≥7/6 +μ₁, thenwhereandAs applications of Theorem 1.1, we'll consider the Priestly-Chao estimate and Gasser-Muller estimate.I. Priestly-Chao estimateAssume that g : [0,1]→R a bounded function, andWhere h_n is a positive constants converging to 0 and nh_n→∞, and the design pointssatisfyAssumption (A4) (1) There exist positive constants c₁ and c₂ such that c₁n≤x_n,i - x_n,i-1≤c₂n^-1, for i = 1,2,... n. (2) K(x) is a continuous and bounded pdf. There exists a majorcant H(x) which is bounded, symmetric, nonincreasing in [0,∞) and integrable over R, such that K(x)≤H(x),x∈R. (3) p and q satisfy (3), and as n→∞Theorem 2.1 If Assumptions (A1),(A2)(2) and (A4) hold, thenCorollary 2.1 Under the conditions of Theorem 2.2, if (4) is replaced byfor 1/2 <σ< 10/13, andσ= 2/3, then2. Gasser-Muller estimateAssume that g : [0,1]→R a bounded function, and Where h_n is a positive constants converging to 0 and nh_n→∞, and the design points satisfy 0 = x_n0≤x(n1)≤…≤x_nn = 1.Theorem 2.2 If Assumptions (A1),(A2)(2) and (A4) are satisfied, thenCorollary 2.2 Under the conditions of Theorem 2.2, if (4) is replaced byfor 1/2 <σ< 10/13, andδ= 2/3, then...