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.