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We generalize the { ${\rm M}$-estimator} put forward by Catoni in his seminalpaper [C12] to the case in which samples can have finite $\alpha$-th momentwith $\alpha \in (1,2)$ rather than finite variance, our approach is byslightly modifying the influence function $\varphi$ therein. The choice of thenew influence function is inspired by the Taylor-like expansion developed in[C-N-X]. We obtain a deviation bound of the estimator, as $\alpha \rightarrow2$, this bound is the same as that in [C12]. Experiment shows that ourgeneralized ${\rm M}$-estimator performs better than the empirical meanestimator, the smaller the $\alpha$ is, the better the performance will be. Asan application, we study an $\ell_{1}$ regression considered by Zhang et al.[Z-Z] who assumed that samples have finite variance, and relax their assumptionto be finite {$\alpha$-th} moment with $\alpha \in (1,2)$.

A generalized Catoni's ${\rm M}$-estimator under finite {$\alpha$-th moment assumption} with $\alpha \in (1,2)$