  This article proposes a new method of truncated estimation to estimate the
tail index $\alpha$ of the extremely heavy-tailed distribution with infinite
mean or variance. We not only present two truncated estimators $\hat{\alpha}$
and $\hat{\alpha}^{\prime}$ for estimating $\alpha$ ($0<\alpha \leq 1$) and
$\alpha$ ($1<\alpha \leq 2$) respectively, but also prove their asymptotic
statistical properties. The numerical simulation results comparing the six
known estimators in estimating error, the Type I Error and the power of
estimator show that the performance of the two new truncated estimators is
quite good on the whole.
