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

We assume T ₁, …, T _n are i.i.d. data sampled from distribution function F with density function f and C ₁, …, C _n are i.i.d. data sampled from distribution function G. Observed data consists of pairs (X _i , δ _i ), i = 1, …, n, where X _i = min{T _i ,C _i }, δ _i = I(T _i ? C _i ), I(A) denotes the indicator function of the set A. Based on the right censored data {X _i , δ _i }, 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 λ _g -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 f _n 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.