We consider a type of heavy random censoring where the number of uncensored observations still tends to infinity. Under natural conditions the life distribution can be locally analyzed by generalizing tail empirical processes to the heavily censored case. A uniform central limit theorem for the tail product-limit process and the tail empirical cumulative hazard process is established. Statistical applications include a local confidence band for the cumulative life distribution and a test concerning the value of its density at the origin.
Key words: heavy censoring, tail product-limit and tail empirical cumulative hazard process, uniform central limit theorem.