Abstract：In this paper , a uniform stro ng representation is obtained for the product-limit estimator F n(x)under random left
truncation data w ith improved erro r bounds.Then w e study the asymptotic proper ties of the kernel density estimation via this strong representation .The asympto tic optimal bandwidth choice and the bound of the MSE are the same as those in comple te sample case.
陈钰芬. 随机截断下PL 估计的强表示式以及密度函数的核估计[J]. 浙江大学学报（理学版）, 1998, 25(3): 12-18.
Chen Yufen. Strong Representation of the Product-Limit Festimator and Kernel Density Estimation Via Strong Representation for Truncated Data. Journal of ZheJIang University(Science Edition), 1998, 25(3): 12-18.
