Uniform Consistency of the Kernel Conditional Kaplan-Meier Estimate

We consider a class of nonparametric regression estimates introduced by Beran to estimate conditional survival functions in the presence of right censoring. An exponential probability bound for the tails of distributions of kernel estimates of conditional survival functions is derived. This inequality is next used to prove weak and strong uniform consistency results. The developments rest on sharp exponential bounds for the oscillation modulus of multivariate empirical processes obtained by Stute.