Asymptotic properties of plug-in level set estimators for right censored data

Abstract

We assume T1, …, Tn are i.i.d. data sampled from distribution function F with density function f and C1, …, Cn are i.i.d. data sampled from distribution function G. Observed data consists of pairs (Xi, δi), i = 1, …, n, where Xi = min{Ti,Ci}, δi = I(Ti ⩽ Ci), I(A) denotes the indicator function of the set A. Based on the right censored data {Xi, δi}, i = 1, …,n, we consider the problem of estimating the level set {f ⩾ c} of an unknown one-dimensional density function f and study the asymptotic behavior of the plug-in level set estimators. Under some regularity conditions, we establish the asymptotic normality and the exact convergence rate of the λg-measure of the symmetric difference between the level set {f ⩾ c} and its plug-in estimator {fn ⩾ c}, where f is the density function of F, and fn is a kernel-type density estimator of f. Simulation studies demonstrate that the proposed method is feasible. Illustration with a real data example is also provided.