ASYMPTOTIC REPRESENTATIONS FOR KERNEL DENSITY AND HAZARD FUNCTION ESTIMATORS WITH LEFT TRUNCATION
Yong Zhou · 1999
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. In many survival studies a subject may not be included in the study if the time origin of its lifetime, called the onset time, precedes the starting time of the study. Such subjects are called left truncated. In this paper we study kernel estimators of density and hazard functions based on the product-limit estima- tor (Lynden-Bell (1971)) when the data is subject to random truncation. Let (Xi ,Y i), 1 ≤ i ≤ N , be a sequence of independent identically distributed random vectors in the plane such that Xi is independent of Yi. The marginal distribution functions of Xi and Yi are given by F (t )= P (X ≤ t )a ndG(t )= P (Y ≤ t), respectively. Let f denote the density function of F. In the random left truncation model one observes only those pairs (Xi ,Y i )f or which Xi ≥ Yi but nothing is observed otherwise. Left truncation is a frequent cause of incomplete data. It may occur if the time origin, X 0 , of the lifetime precedes the time origin of the study, X 1 . To be precise, when Y � = X 1 −X 0 >X , where X is the lifetime of interest, the case is not observed at all (we do not even know its existence). An important example of such a model arises in the analysis of survival data of patients infected by the AIDS virus from contaminated blood transfusions (Chen, Chao and Lo (1995) and Lagakos, Barraj and De Gruttola (1988)). A feature of HIV (AIDS) development is the induction period between infection with the AIDS virus and the onset of clinical AIDS. The data collected on persons infected by contaminated blood transfusions provide a unique source