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.

翻译：本文将充分性概念推广至最大似然估计以外的估计问题。具体而言，我们考虑了基于Jones和Basu等提出的似然函数的估计问题，这些似然函数在基于距离的稳健推断方法中广为应用。首先，我们刻画了那些相对于这些似然函数恒具有固定数量充分统计量（与样本量无关）的概率分布。这些分布是通常指数族的幂律推广，并以学生分布作为特例。随后，我们拓展了极小充分统计量的概念，并针对这些幂律族计算了该统计量。最后，我们建立了用于寻找幂律族最优估计量的Rao-Blackwell型定理，这有助于建立幂律族的Cramér-Rao型下界。