  Based on suitable left-truncated or censored data, two flexible classes of
$M$-estimations of Weibull tail coefficient are proposed with two additional
parameters bounding the impact of extreme contamination. Asymptotic normality
with $\sqrt {n}$-rate of convergence is obtained. Its robustness is discussed
via its asymptotic relative efficiency and influence function. It is further
demonstrated by a small scale of simulations and an empirical study on CRIX.
