Uniform-in-bandwidth kernel estimation for censored data

• Lifetimes are subject to right random censoring.
• We get a Strassen type functional limit law for the Kaplan–Meier empirical process.
• We derive limit laws for kernel estimators of lifetime density and hazard rate.
• Uniform-in-bandwidth results permit to use kernel estimators with random bandwidth.
• We construct uniform asymptotic certainty bands for the lifetime density.

We present a sharp uniform-in-bandwidth functional limit law for the increments of the Kaplan–Meier empirical process based upon right-censored random data. We apply this result to obtain limit laws for nonparametric kernel estimators of local functionals of lifetime densities, which are uniform with respect to the choices of bandwidth and kernel. These are established in the framework of convergence in probability, and we allow the bandwidth to vary within the complete range for which the estimators are consistent. We provide explicit values for the asymptotic limiting constant for the sup-norm of the estimation random error.