Regression Function Kernel Estimation Based on Synthetic Data Under Random Censorship

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.