摘要
本文考虑截尾数据情况下非参数回归函数m(x)=E(Y|x)的估计。具体地讲,我们面对的是这样的数学模型:T是与(X,Y)独立的随机变量,我们观测到的不是Y本身,而是Z=min(Y,T)及δ=[Y≤T]。今有训练样本{(X_i,Z_i,δ_i)}_(i-1)及当前样本(X,z,δ),记ξ_i(·)=[z_i≥·], N^+(·)=sum from i=1 to n ξ_i(·), V_n(·)=multiply from i=1 to n{1+N^+(z_i)/2+N^+(z_i)}~[δ_i=_i<0], U_n(·)=sum from i=1 to n Wnt(x)ξ_i(·), 令 m_n(x)=integral from 0 to u_n U_n(y)|V_n(y)dy, 其中u_n=F_2^(-1)(n^(-a)),0<α<1/2为一实常数,F_2(·)=P(Y≥·)为Y的(右侧)分布函数。在权函数{W_(ni)(x)}_(i=1)~n及(X,Y,T)的分布函数满足一组条件下,我们证明了m_n(x)为m(x)的强相合估计,即:m_n(x)→m(x),a.s.(n→+∞).
Let (X_1, Y_1), …, (X_n, Y_n) be i.i.d. R^d×R^1 random vectoos, E|Y|<+∞. then m(x)=E(Y|x) is called a regression function, Now, T_1, …, T_n be i.i.d. samples of random variable T, independent of {(X_i, Y_i)}_(i=1)~n. Set F(x, y)=P(X≥x, Y≥y), G(t)=P(T≥t), both F and G are unknown continuous survival functions. Based on obserations Z_i=min(X_i, Y_i) and δ_i=[X_i≤Y_i] only, we proposed an estimate m_n(x) of m(x). Under some conditions it is shown that m_n(x)→m(x), a. s. (m→+∞).
出处
《应用概率统计》
CSCD
北大核心
1990年第1期57-61,共5页
Chinese Journal of Applied Probability and Statistics
