幂律分布的广义 Fisher-Darmois-Koopman-Pitman 定理与 Rao-Blackwell 型估计量
Generalized Fisher-Darmois-Koopman-Pitman Theorem and Rao-Blackwell Type Estimators for Power-Law Distributions
AI总结：
本文将充分性推广至基于距离的稳健似然，刻画具有固定维充分统计量的幂律分布族，给出最小充分统计量，并建立 Rao-Blackwell 型定理及 Cramér-Rao 型下界。
摘要：
本文将估计问题中的充分性概念推广到最大似然之外。具体而言，我们考虑基于 Jones 等人和 Basu 等人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很常用。我们首先刻画了相对于这些似然函数始终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，并将 Student 分布作为特例包含在内。随后，我们扩展了最小充分统计量的概念，并为这些幂律族计算了该统计量。最后，我们建立了一个 Rao-Blackwell 型定理，用于寻找幂律族的最佳估计量。这帮助我们建立了幂律族的 Cramér-Rao 型下界。
英文摘要：
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 Cramér-Rao-type lower bounds for power-law families.
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