We initiate a nonparametric large sample estimation theory of total time on test transforms and LORENZ curves under random censorship from the right. We introduce appropriate functional estimators for these functions based on teh product-limit estimator and its quantile function and develop strong uniform consistency results and weak convergence theorems in the form of weak approximations by sequences of appropriate copies of the limiting GAUSSian processes. Our main technical tools are the CHIBISOV-O'REILLY theorems for the uniform product-limit and product-limit quantile processes which we establish here. These are of independent interest.