The kernel function of kernel estimator for regression function is often defined with compact support, the moment of Y is bigger than one. The convergence rate of the improved kernel estimator is discussed under the improved kernel function. The result is also discussed under random censorship. The class of applicable kernels include those having unbounded support and even not integral .The condition of moment of Y is discarded under the improved kernel estimator. The results are wholly the same as usual and generalize the relative results of the paper .