截尾样本下回归函数改良核估计的强相合性

Strong Consistency of Improved Kernel Estimate of Regression Function with Censored Data

摘要: 设（X_i,Y_i）,i=1，…，n是从取值于R^d×R¹的随机向量（X，Y）中抽取的i.i.d.样本,Ｅ(|Y|)＜∞，而以m(x)＝Ｅ(Y|X=x)表示回归函数．在截尾情况下，观察到的不是诸Y_i本身，而是Z_i=\min \left(Y_i, T_i\right)及\delta_i=I\left(Y_i \leq T_i\right) 其中T_i是与（X_i,Y_i）独立的随机变量，,i=1，…，n当 T的分布未知时，在一定条件下，得到了回归函数改良估计的强合性。

Abstract: Let （X, Y） be a R^d×R¹-Valued random vector with E（|Y|）＜ ∞ and m（x） = E（Y|X = x） be the regression of Y with respect to X. Suppose that （X_i,Y_i）,i=1，…，n are i.i.d. samples drawn from （X, Y）, it is desired to estimate m（x） based on these samples. In this paper we discuss the case that Y_i are censored by random variables T_i. It means that we can only observe Z_i=\min \left(Y_i, T_i\right) and \delta_i=I\left(Y_i \leq T_i\right). We always suppose that T_i i.i.d. and independent of （X_i,Y_i）. We obtain strong cosistency of regression function.