A generalized Catoni’s M-estimator under finite α-th moment assumption with α∈(1,2)

Abstract

We generalize Catoni’s M-estimator, put forward in [3] by Catoni under finite variance assumption, to the case in which distributions can have finite α-th moment with $\mathit{\alpha }\in (1,2)$. Our approach, inspired by the Taylor-like expansion developed in [4], is via slightly modifying the influence function φ in [3]. A deviation bound is established for this generalized estimator, and coincides with that in [3] as $\mathit{\alpha }\uparrow 2$. Experiment shows that our M-estimator performs better than the empirical mean, the smaller the α is, the better the performance will be. As an application, we study an ${\ell }_1$ regression considered by Zhang et al. [19], who assumed that samples have finite variance, under finite α-th moment assumption with $\mathit{\alpha }\in (1,2)$.