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

TL;DR

This paper extends classical statistical theorems to power-law distributions, characterizing sufficient statistics and optimal estimators within this broader family, including robust inference methods.

Contribution

It generalizes the Fisher-Darmois-Koopman-Pitman theorem and Rao-Blackwell estimators to power-law distributions, expanding the theoretical framework beyond exponential families.

Findings

Characterized distributions with fixed number of sufficient statistics for power-law families.

Computed minimal sufficient statistics for these families.

Established Rao-Blackwell-type theorem and Cramér-Rao bounds for power-law distributions.

Abstract

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.