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This paper generalizes the notion of sufficiency for estimation problemsbeyond maximum likelihood. In particular, we consider estimation problems basedon Jones et al. and Basu et al. likelihood functions that are popular amongdistance-based robust inference methods. We first characterize the probabilitydistributions that always have a fixed number of sufficient statistics withrespect to these likelihood functions. These distributions are power-lawextensions of the usual exponential family and contain Student distributions asspecial case. We then extend the notion of minimal sufficient statistics andcompute it for these power-law families. Finally, we establish a Rao-Blackwelltype theorem for finding best estimators for a power-law family. This enablesus to derive certain generalized Cram\'er-Rao lower bounds for power-lawfamilies.

Generalized Fisher-Darmois-Koopman-Pitman Theorem and Rao-Blackwell Type Estimators for Power-Law Distributions