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

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）, em= 1,..., n, where Xi = min{Ti,Ci}, δi = I（Ti 6 Ci）, I（A） denotes the indicator function of the set A. Based on the right censored data {Xi, δi}, em=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.

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 theλg-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.