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

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.