  A model for competing (resp. complementary) risks survival data where the
failure time can be left (resp. right) censored is proposed. Product-limit
estimators for the survival functions of the individual risks are derived. We
deduce the strong convergence of our estimators on the whole real half-line
without any additional assumptions and their asymptotic normality under
conditions concerning only the observed distribution. When the observations are
generated according to the double censoring model introduced by Turnbull, the
product-limit estimators represent upper and lower bounds for Turnbull's
estimator.
