随机删失场合基于Synthetic Data的回归函数的核估计及强相合性

Regression Function Kernel Estimation Based on Synthetic Data Under Random Censorship

摘要: 设（X₁,Y₁）,…,（X_n,Y_n）是来自总体（X,Y）的取值于R^d×R上的i.i.d.随机向量,m(x)=E(Y \mid X=x)是未知的非参数回归函数,Y_i被随机变量T_i删失,只能观察到Z_i=\min \left\Y_i, T_i\right\, \delta_i=\leftY_i \leq T_i\right, i=1, \cdots, n,本文分别在T_i的分布函数已知和未知的情形下,利用Leurgans等人提出的Synthetic data方法获得新的数据Y_i^*与Y_i^**,考虑了m（x）的核估计并且证明了其强相合性。

Abstract: Let(X₁,Y₁）,…,（X_n,Y_n） be i.i.d. Rd×R random vectors coming from population （X,Y）, and m(x)=E(Y \mid X=x) be a unknown nonparametric regression function. Now, Y_i are randomly censored by T_i , where T_i are i.i.d. samples of random variable T, independent of （X_i, Y_i）. We can only observe Z_i=\min \left\Y_i, T_i\right\, \delta_i=\leftY_iY_i^* and Y_i^** using the synthetic data method proposed by Leurgans, etc., and proposes the kernel estimates M^*（x） and m^**（x） of m（x）, under some conditions, these estimates are shown strongly consistent.