Weak and strong uniform consistency rates of kernel density estimates for randomly censored data

Let X 1 ,., X n , be i.i.d. random variables with distribution function F , and let Y 1 ,.,.,Y n be i.i.d. with distribution function G. For i = 1, 2,.,., n set δ i , = 1 if X i ≤ Y i , and 0 otherwise, and X i , = min{X i , K i }. A kernel‐type density estimate of f , the density function of F w.r.t. Lebesgue measure on the Borel o‐field, based on the censored data (δ i , X i ), i = 1,.,., n , is considered. Weak and strong uniform consistency properties over the whole real line are studied. Rates of convergence results are established under higher‐order differentiability assumption on f . A procedure for relaxing such assumptions is also proposed.