This paper generalizes the notion of sufficiency for estimation problems beyond maximum likelihood. In particular, we consider estimation problems based on Jones et al. and Basu et al. likelihood functions that are popular among distance-based robust inference methods. We first characterize the probability distributions that always have a fixed number of sufficient statistics (independent of sample size) with respect to these likelihood functions. These distributions are power-law extensions of the usual exponential family and contain Student distributions as a special case. We then extend the notion of minimal sufficient statistics and compute it for these power-law families. Finally, we establish a Rao-Blackwell-type theorem for finding the best estimators for a power-law family. This helps us establish Cramér-Rao-type lower bounds for power-law families.

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.

In [12], it was shown that a generalized maximum likelihood estimation problem on a (canonical) α-powerlaw model (M(α)-family) can be solved by solving a system of linear equations. This was due to an orthogonality relationship between the M(α)-family and a linear family with respect to the relative α-entropy (or the Iα-divergence). Relative α-entropy is a generalization of the usual relative entropy (or the Kullback-Leibler divergence). M(α)-family is a generalization of the usual exponential family. In this paper, we first generalize the M(α)-family including the multivariate, continuous case and show that the Student-t distributions fall in this family. We then extend the above stated result of [12] to the general M(α)-family. Finally we apply this result to the Student-t distribution and find generalized estimators for its parameters.