Asymptotic Properties Of Estimates Of Nonparametric Regression Models Based On Negatively Associated Sequences

Asymptotic properties of estimates of nonparametric regression models based on negatively associated sequencesNonparametric regression model:Y_ni=g(x_ni)+ε_ni=1,2,…, n.where g is an unknown regression function, x_ni are known fixed design points, andε_niare random errors ,considering weighted regression estimator:g_n(x)=∑_i=1ⁿW_ni(x)Y_ni where W_niare weighted regression estimator. g_n(x)as an estimator of g.We study the above nonparametric regression problem under negative association. A finite family of random variables {x_i, 1≤i≤n} are negative associated (NA),if for every pair of disjoint subsets A and B of {1,2,…, n}, we haveCov(f₁(x_i,i∈a),f₂(x_j,j∈b))≤0 Whenever f₁ and f₂ are coordinatewise increasing and such that the covariance exists. An infinite family of random variables is NA if every finite subfamily is NA.The sample of the paper (x_ni, y_ni),1≤i≤n derive from the regression model,Y_ni=g(x_ni)+ε_ni, i=1,…,ngis an unknown real value regression function,assuming g(x) is bounded on a compact set Ain R^p,X_ni are known fixed design points from A,and {ε_ni,i≥1}from a sequence of zero mean NA random errors. We shall consider the following weighted regression estimator of g:g_n(x)=sum from i=1 to n W_ni(x)Y_ni,x∈A(?)R^PW_niare of the form W_ni(x)=W_ni(x, x_ni,…, x_nn)For any function g(x), we use c(g) to denote all continuity points of the function g(x)on A.In this paper, we at first discussed the situation of g_n(x)Lp convergent to g(x),mainly as following:Under the assumption:(a)sum from i=1 to n W_ni{x)→1, as n→∞;(b)sum from i=1 to n|W_ni(x)|≤C,for all n(c)sum from i=1 to n|W_ni(x)|I(‖X_ni-X‖>a)→0 as n→∞for all a>0, we get,Theorem 1 Under assumption (a)-(c) , If sup_i E|ε_ni|^p<∞, for some p>1, and there exist 1 < s < min 2,p,such that sum from i=1 to n|W_ni(x)|^s→0, as n→∞, thenx∈(?)(g),E|g_n(x)-g(x)|^p→0当n→∞instead of the assumption(a)-(c), we offer uniform version of (a)-(c).(a')(?)|sum from i=1 to n W_ni(x)-1|=o(1);(b')(?) sum from i=1 to n|W_ni(x)|≤C for all n;(c')(?) sum from i=1 to n|W_ni(x)|I(‖x_ni-x‖>a)=o(1)对a>0;(d')There existsα>0 and set A partition A₁⁽ⁿ⁾,A₂⁽ⁿ⁾…, A_c1n^α+c2⁽ⁿ⁾, such that for any u, v∈A_k⁽ⁿ⁾, 1≤k≤c₁n^α+c₂,as nbig enough,W_ni(u)-W_ni(v),1≤i≤n. we get the results as following.Theorem 2 Under assumption (a')-(d'), and sup_i E|ε_ni|^p<∞for p>1, if there exist 1 < s≤min{2,p},such that sup_∈A sum from i=1 to n|W_ni(x)|^s→0, as n→∞, then(?) sup E|g_n(x)-g(x)|^p=0Theorem 3 Under assumption sup_iE|ε_ni|^p<∞, for some p > 1 and let k_n=β(n^β).If 0<β<1, then(?)E|(g|⁾_n(x)-g(x)|^p=0We study nonparametric estimator g_n(x)under the conditions (a)-(b) in the middle part of paper, if the weighted function W_ni(x)and random errorsε_nirespectively satisfied various conditions, the strong convergence of g_n(x)Theorem 4 Suppose that assumption (a)-(c) is satisfied , if sup_i E|ε_ni|^p<∞for p>1, and there exist s∈(1/p, 1) such that sup_i|W_ni(x)|=O(n^-s), then there exist (?)x∈c(g),g_n(x)→g(x)a.s. n→∞Theorem 5 Assume that conditions (a')-(d') hold,sup_i E|ε_ni|^p<∞,for p>1, if there exist s∈(1/p, 1) such that sup_x∈A sup_i |W_ni(x)|=O(n^-s), then(?)|g_n(x)-g(x)|=0.a.sTheorem 6 Suppose the assumption sup_i E|ε_ni|^p<∞is satisfied, for some p>1, and there exist k_n=O(n^β) ifβ∈(1/p, 1),then(?)|(g|⁾_n(x)-g(x)|=0.a.sTheorem 7 Assume that conditions (a)-(c)hold,and there are {ε_ni, i≥1}satisfy that sup_i p(|ε_ni|≥t)=O(1)p(|ε|≥t), (?)t>0, if E|ε|^1+1/s<∞,for some s > 0, and sup_i|W_ni(x)|=O(n^-s), then (?)x∈c(g),g_n(x)→g(x) a.s. n→∞ Theorem 8 Under assumption (a)-(c), if there exist sup_i|ε_ni|<∞a.s. and sup_i|W_ni(x)| log n→0 as n→∞then (?)x∈c(g)g_n(x)→g(x), n→∞At last, we summarize the under the conditions of negative association the asymptotic normality of nonparametric regression estimator g_n(x),for the regression function g(x),letσ_n²(x)-Var(g_n(x)).we need to use the assumption:(A1)ε_ni≥1 is uniformly integrable in L₂ and satisfies:V(u) :=sup_k≥1∑_{j:|k-j|≥u}|Cov(ε_nk,ε_nj|→0, as u→∞(A2)The weight satisfy,sum from i=1 to n W_ni²(x)=O(σ_n²(x)) and max₍1≤i≤n|W_ni(x)|=o(σ_n(x)),Theorem 9 Suppose the assumption (A1)-(A2) is satisfied, we have(g_n(x)-Eg_n(x))/(σ_n(x))→N(0,1)Under the assumption:(B1)Let{Z_n}be a sequence of zero mean NA random variables and satisfy uniformly integrable in L₂ and (?) |Cov(Z_k, Z_j)|→0,as u→∞(B2)For each n,{ε_ni,1≤i≤n}have the same distribution asξ₁,ξ₂…,ξ_n,whereξ_t=sum from j=0 to∞c_jZ_t-j,here{c_j}is a sequence of real numbers with sum from j=0 to∞|c_j|<∞(B3) The weights satisfysum from i=1 to n|W_ni(x)|≤C(?)|W_ni(x)|=O(sum from i=1 to n W_ni²)(x)),sum from i=1 to n W_ni²(x)=O(σ_n²)(x),(?)|W_ni(x)|=o(σ_n(x))we have: Theorem 10 Under the assumption (B1)-(B3),ifσ_n^-1(x)∑_j>n^∞|c_j|→0, then(g_n(x)-Eg_n(x))/(σ_n(x))→N(0,1)...