A note on the integrated square errors of kernel density estimators under random censorship

Randomly censored data consist of i.i.d. pairs of observations (Xi,δi), i=1,…,n. If δi=0, Xi denotes a censored observation, and if δi=1, Xi denotes a survival time, which is the variable of interest. A popular stochastic measure of the distance between the density function f of the survival times and its kernel estimate fn is the integrated square error. In this paper, we apply the technique of strong approximation to establish an asymptotic expansion for the integrated square error of the kernel density estimate fn.