  We address the estimation of conditional quantiles when the covariate is
functional and when the order of the quantiles converges to one as the sample
size increases. In a first time, we investigate to what extent these large
conditional quantiles can still be estimated through a functional kernel
estimator of the conditional survival function. Sufficient conditions on the
rate of convergence of their order to one are provided to obtain asymptotically
Gaussian distributed estimators. In a second time, basing on these result, a
functional Weissman estimator is derived, permitting to estimate large
conditional quantiles of arbitrary large order. These results are illustrated
on finite sample situations.
