  We revisit the estimation of the extreme value index for randomly censored
data from a heavy tailed distribution. We introduce a new class of estimators
which encompasses earlier proposals given in Worms and Worms (2014) and
Beirlant et al. (2018), which were shown to have good bias properties compared
with the pseudo maximum likelihood estimator proposed in Beirlant et al. (2007)
and Einmahl et al. (2008). However the asymptotic normality of the type of
estimators first proposed in Worms and Worms (2014) was still lacking, in the
random threshold case. We derive an asymptotic representation and the
asymptotic normality of the larger class of estimators and consider their
finite sample behaviour. Special attention is paid to the case of heavy
censoring, i.e. where the amount of censoring in the tail is at least 50\%. We
obtain the asymptotic normality with a classical $\sqrt{k}$ rate where $k$
denotes the number of top data used in the estimation, depending on the degree
of censoring.
