  We introduce a kernel estimator, to the tail index of a right-censored
Pareto-type distribution, that generalizes Worms's one (Worms and Worms,
2014)in terms of weight coefficients. Under some regularity conditions, the
asymptotic normality of the proposed estimator is established. In the framework
of the second-order condition, we derive an asymptotically bias-reduced version
to the new estimator. Through a simulation study, we conclude that one of the
main features of the proposed kernel estimator is its smoothness contrary to
Worms's one, which behaves, rather erratically, as a function of the number of
largest extreme values. As expected, the bias significantly decreases compared
to that of the non-smoothed estimator with however a slight increase in the
mean squared error.
