We propose a generalized notion of principle of sufficiency when the underlying inference method is not necessarily maximum likelihood. This notion is based on certain generalized likelihood functions that arise in robust inference problems. Particularly, in this paper, we consider the Basu et al. estimation [1]. We identify the specific form of the probability distributions that have a fixed number of sufficient statistics with respect to this estimation. These distributions are power-law in nature and Student distributions are a part of this family.