  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 Cram\'er-Rao-type
lower bounds for power-law families.
