  We generalize the { ${\rm M}$-estimator} put forward by Catoni in his seminal
paper [C12] to the case in which samples can have finite $\alpha$-th moment
with $\alpha \in (1,2)$ rather than finite variance, our approach is by
slightly modifying the influence function $\varphi$ therein. The choice of the
new influence function is inspired by the Taylor-like expansion developed in
[C-N-X]. We obtain a deviation bound of the estimator, as $\alpha \rightarrow
2$, this bound is the same as that in [C12]. Experiment shows that our
generalized ${\rm M}$-estimator performs better than the empirical mean
estimator, the smaller the $\alpha$ is, the better the performance will be. As
an application, we study an $\ell_{1}$ regression considered by Zhang et al.
[Z-Z] who assumed that samples have finite variance, and relax their assumption
to be finite {$\alpha$-th} moment with $\alpha \in (1,2)$.
