Asymptotic representations for kernel density and hazard function estimators with left truncation

Kernel estimators of the density function and the hazard function based on the product-limit estimator are considered when the data are subject to left truncation. Several asymptotic uniformly strong and weak representations for these estimators are established. Making use of these results we obtain the large sample properties of the kernel density and hazard function estimators. The results can be extended to the case of left truncated and right censored data.