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

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 with respect to these likelihood functions. These distributions are power-law extensions of the usual exponential family and contain Student distributions as 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 best estimators for a power-law family. This enables us to derive certain generalized Cram\'er-Rao lower bounds for power-law families.